Divisibility rule - Knowledge

2339:. That is, the divisibility of any number by seven can be tested by first separating the number into digit pairs, and then applying the algorithm on three digit pairs (six digits). When the number is smaller than six digits, then fill zero's to the right side until there are six digits. When the number is larger than six digits, then repeat the cycle on the next six digit group and then add the results. Repeat the algorithm until the result is a small number. The original number is divisible by seven if and only if the number obtained using this algorithm is divisible by seven. This method is especially suitable for large numbers. 2151:, as the multiplier. Start on the right. Multiply by 5, add the product to the next digit to the left. Set down that result on a line below that digit. Repeat that method of multiplying the units digit by five and adding that product to the number of tens. Add the result to the next digit to the left. Write down that result below the digit. Continue to the end. If the result is zero or a multiple of seven, then yes, the number is divisible by seven. Otherwise, it is not. This follows the Vedic ideal, one-line notation. 3591:. This implies that a number is divisible by 13 iff removing the first digit and subtracting 3 times that digit from the new first digit yields a number divisible by 13. We also have the rule that 10 x + y is divisible iff x + 4 y is divisible by 13. For example, to test the divisibility of 1761 by 13 we can reduce this to the divisibility of 461 by the first rule. Using the second rule, this reduces to the divisibility of 50, and doing that again yields 5. So, 1761 is not divisible by 13. 2451: 3935: 1836:. One must multiply the leftmost digit of the original number by 3, add the next digit, take the remainder when divided by 7, and continue from the beginning: multiply by 3, add the next digit, etc. For example, the number 371: 3×3 + 7 = 16 remainder 2, and 2×3 + 1 = 7. This method can be used to find the remainder of division by 7. 4412: 2166:

The Pohlman–Mass method provides a quick solution that can determine if most integers are divisible by seven in three steps or less. This method could be useful in a mathematics competition such as MATHCOUNTS, where time is a factor to determine the solution without a calculator in the Sprint Round.

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The rules given below transform a given number into a generally smaller number, while preserving divisibility by the divisor of interest. Therefore, unless otherwise noted, the resulting number should be evaluated for divisibility by the same divisor. In some cases the process can be iterated until

2199:

Step B: If the integer is between 1001 and one million, find a repeating pattern of 1, 2, or 3 digits that forms a 6-digit number that is close to the integer (leading zeros are allowed and can help you visualize the pattern). If the positive difference is less than 1000, apply Step A. This can be

1949:

Repeat the procedure until you have a recognizable multiple of 7, or to make sure, a number between 0 and 6. So, starting from 21 (which is a recognizable multiple of 7), take the first digit (2) and convert it into the following in the sequence above: 2 becomes 6. Then add this to the second digit:

1554:

The basic rule for divisibility by 4 is that if the number formed by the last two digits in a number is divisible by 4, the original number is divisible by 4; this is because 100 is divisible by 4 and so adding hundreds, thousands, etc. is simply adding another number that is divisible by 4. If any

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a rule for divisibility by any integer relatively prime to 10 (including 33 and 39; see the table below). This is why the last divisibility condition in the tables above and below for any number relatively prime to 10 has the same kind of form (add or subtract some multiple of the last digit from

3638:

the prime under consideration (does not work for 2 or 5) and use that as a multiplier to make the divisibility of the original number by that prime depend on the divisibility of the new (usually smaller) number by the same prime. Using 31 as an example, since 10 × (−3) = −30 = 1 mod 31, we get the

3625:

divisor may also have a rule formed using the same procedure as for a prime divisor, given below, with the caveat that the manipulations involved may not introduce any factor which is present in the divisor. For instance, one cannot make a rule for 14 that involves multiplying the equation by 7.

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Multiply the remainders with the appropriate multiplier from the sequence 1, 2, 4, 1, 2, 4, ... : the remainder from the digit pair consisting of ones place and tens place should be multiplied by 1, hundreds and thousands by 2, ten thousands and hundred thousands by 4, million and ten million

2206:

The fact that 999,999 is a multiple of 7 can be used for determining divisibility of integers larger than one million by reducing the integer to a 6-digit number that can be determined using Step B. This can be done easily by adding the digits left of the first six to the last six and follow with

1839:

A more complicated algorithm for testing divisibility by 7 uses the fact that 10 ≡ 1, 10 ≡ 3, 10 ≡ 2, 10 ≡ 6, 10 ≡ 4, 10 ≡ 5, 10 ≡ 1, ... (mod 7). Take each digit of the number (371) in reverse order (173), multiplying

1797:

is divisible by 7. In other words, subtract twice the last digit from the number formed by the remaining digits. Continue to do this until a number is obtained for which it is known whether it is divisible by 7. The original number is divisible by 7 if and only if the number obtained using this

2210:

Step C: If the integer is larger than one million, subtract the nearest multiple of 999,999 and then apply Step B. For even larger numbers, use larger sets such as 12-digits (999,999,999,999) and so on. Then, break the integer into a smaller number that can be solved using Step B. For example:

2071:

What this procedure does, as explained above for most divisibility rules, is simply subtract little by little multiples of 7 from the original number until reaching a number that is small enough for us to remember whether it is a multiple of 7. If 1 becomes a 3 in the following decimal position,

1957:

If at any point the first digit is 8 or 9, these become 1 or 2, respectively. But if it is a 7 it should become 0, only if no other digits follow. Otherwise, it should simply be dropped. This is because that 7 would have become 0, and numbers with at least two digits before the decimal dot do not

1661:

If the last digit in the number is 5, then the result will be the remaining digits multiplied by two, plus one. For example, the number 125 ends in a 5, so take the remaining digits (12), multiply them by two (12 × 2 = 24), then add one (24 + 1 = 25). The result is the same as the result of 125

1464:

First, take any number (for this example it will be 376) and note the last digit in the number, discarding the other digits. Then take that digit (6) while ignoring the rest of the number and determine if it is divisible by 2. If it is divisible by 2, then the original number is divisible by 2.

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First, take any number (for this example it will be 492) and add together each digit in the number (4 + 9 + 2 = 15). Then take that sum (15) and determine if it is divisible by 3. The original number is divisible by 3 (or 9) if and only if the sum of its digits is divisible by 3 (or 9).

145:, we can separately test for divisibility by each prime to its appropriate power. For example, testing divisibility by 24 (24 = 8×3 = 2×3) is equivalent to testing divisibility by 8 (2) and 3 simultaneously, thus we need only show divisibility by 8 and by 3 to prove divisibility by 24. 1756:

If the final digit is even the number is divisible by two, and thus may be divisible by 6. If it is divisible by 2 continue by adding the digits of the original number and checking if that sum is a multiple of 3. Any number which is both a multiple of 2 and of 3 is a multiple of 6.

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If the last digit in the number is 0, then the result will be the remaining digits multiplied by 2. For example, the number 40 ends in a zero, so take the remaining digits (4) and multiply that by two (4 × 2 = 8). The result is the same as the result of 40 divided by 5(40/5 = 8).

631:

Compute the remainder of each digit pair (from right to left) when divided by 7. Multiply the rightmost remainder by 1, the next to the left by 2 and the next by 4, repeating the pattern for digit pairs beyond the hundred-thousands place. Adding the results gives a multiple of 7.

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For example, in base 10, the factors of 10 include 2, 5, and 10. Therefore, divisibility by 2, 5, and 10 only depend on whether the last 1 digit is divisible by those divisors. The factors of 10 include 4 and 25, and divisibility by those only depend on the last 2 digits.

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Similarly, when you turn a 3 into a 2 in the following decimal position, you are turning 30×10 into 2×10, which is the same as subtracting 30×10−28×10, and this is again subtracting a multiple of 7. The same reason applies for all the remaining conversions:

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procedure is divisible by 7. For example, the number 371: 37 − (2×1) = 37 − 2 = 35; 3 − (2 × 5) = 3 − 10 = −7; thus, since −7 is divisible by 7, 371 is divisible by 7.

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Multiply the right most digit by the left most digit in the sequence and multiply the second right most digit by the second left most digit in the sequence and so on and so for. Next, compute the sum of all the values and take the modulus of 7.

622:

Multiply each digit (from right to left) by the digit in the corresponding position in this pattern (from left to right): 1, 3, 2, −1, −3, −2 (repeating for digits beyond the hundred-thousands place). Adding the results gives a multiple of 7.

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to 1 modulo 3. Since two things that are congruent modulo 3 are either both divisible by 3 or both not, we can interchange values that are congruent modulo 3. So, in a number such as the following, we can replace all the powers of 10 by 1:

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Also, one can simply divide the number by 2, and then check the result to find if it is divisible by 2. If it is, the original number is divisible by 4. In addition, the result of this test is the same as the original number divided by 4.

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Find remainder of 1036125837 divided by 7 1×3 + 0 = 3 3×3 + 3 = 12 remainder 5 5×3 + 6 = 21 remainder 0 0×3 + 1 = 1 1×3 + 2 = 5 5×3 + 5 = 20 remainder 6 6×3 + 8 = 26 remainder 5 5×3 + 3 = 18 remainder 4 4×3 + 7 = 19 remainder 5 Answer is 5

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Because 1001 is divisible by seven, an interesting pattern develops for repeating sets of 1, 2, or 3 digits that form 6-digit numbers (leading zeros are allowed) in that all such numbers are divisible by seven. For example:

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This allows adding and subtracting alternating sets of three digits to determine divisibility by seven. Understanding these patterns allows you to quickly calculate divisibility of seven as seen in the following examples:

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This method can be simplified by removing the need to multiply. All it would take with this simplification is to memorize the sequence above (132645...), and to add and subtract, but always working with one-digit numbers.

5307: 4207: 3930:{\displaystyle D(n)\equiv {\begin{cases}9a+1,&{\mbox{if }}n{\mbox{ = 10a+1}}\\3a+1,&{\mbox{if }}n{\mbox{ = 10a+3}}\\7a+5,&{\mbox{if }}n{\mbox{ = 10a+7}}\\a+1,&{\mbox{if }}n{\mbox{ = 10a+9}}\end{cases}}\ } 2170:

Step A: If the integer is 1000 or less, subtract twice the last digit from the number formed by the remaining digits. If the result is a multiple of seven, then so is the original number (and vice versa). For example:

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Multiply the right most digit of the number with the left most number in the sequence shown above and the second right most digit to the second left most digit of the number in the sequence. The cycle goes on.

1876:= 1 + 21 + 6 = 28). The original number is divisible by 7 if and only if the number obtained using this procedure is divisible by 7 (hence 371 is divisible by 7 since 28 is). 2681: 4780:

The representation of the number may also be multiplied by any number relatively prime to the divisor without changing its divisibility. After observing that 7 divides 21, we can perform the following:

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is divisible by 7. So add five times the last digit to the number formed by the remaining digits, and continue to do this until a number is obtained for which it is known whether it is divisible by 7.

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Using the example above: 16,499,205,854,376 has four of the digits 1, 4 and 7 and four of the digits 2, 5 and 8; Since 4 − 4 = 0 is a multiple of 3, the number 16,499,205,854,376 is divisible by 3.

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Add the result in the previous step (2) to the second digit of the number, and substitute the result for both digits, leaving all remaining digits unmodified: 2 + 0 = 2. So

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Alternatively, one can just add half of the last digit to the penultimate digit (or the remaining number). If that number is an even natural number, the original number is divisible by 4

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For all of the above examples, subtracting the first three digits from the last three results in a multiple of seven. Notice that leading zeros are permitted to form a 6-digit pattern.

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number ends in a two digit number that you know is divisible by 4 (e.g. 24, 04, 08, etc.), then the whole number will be divisible by 4 regardless of what is before the last two digits.

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Most numbers do not divide 9 or 10 evenly, but do divide a higher power of 10 or 10 − 1. In this case the number is still written in powers of 10, but not fully expanded.

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The method is based on the observation that 100 leaves a remainder of 2 when divided by 7. And since we are breaking the number into digit pairs we essentially have powers of 100.

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Is 355,341 divisible by seven? 3 × 3 + 5 = 14 -> remainder 0 -> 0×3 + 5 = 5 -> 5×3 + 3 = 18 -> remainder 4 -> 4×3 + 4 = 16 -> remainder 2 -> 2×3 + 1 = 7 YES

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of the same kind - our choice of addition or subtraction being dictated by arithmetic convenience of the smaller value. In fact, this rule for prime divisors besides 2 and 5 is

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001 001 = 1,001 / 7 = 143 010 010 = 10,010 / 7 = 1,430 011 011 = 11,011 / 7 = 1,573 100 100 = 100,100 / 7 = 14,300 101 101 = 101,101 / 7 = 14,443 110 110 = 110,110 / 7 = 15,730

3540: 5207: 3731:= 10×3+91 = 121; this is divisible by 11 (with quotient 11), so 913 is also divisible by 11. As another example, to determine if 689 = 10×68 + 9 is divisible by 53, find that 4673: 69:

For divisors with multiple rules, the rules are generally ordered first for those appropriate for numbers with many digits, then those useful for numbers with fewer digits.

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Sum the ones digit, 4 times the 10s digit, 4 times the 100s digit, 4 times the 1000s digit, etc. If the result is divisible by 6, so is the original number. (Works because

5368: 4817: 4623: 5014: 4767: 2796: 1521:, the digits will add up to the remainder of the original number if divided by eleven, and numbers are divisible by eleven only if the digit sum is divisible by eleven. 4575:

This applies to divisors that are a factor of a power of 10. This is because sufficiently high powers of the base are multiples of the divisor, and can be eliminated.

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of the original number if it were divided by nine (unless that single digit is nine itself, in which case the number is divisible by nine and the remainder is zero).

2235:

Is 42,341,530 divisible by seven? 42,341,530 -> 341,530 + 42 = 341,572 (Step C) 341,572 − 341,341 = 231 (Step B) 231 -> 23 − (1×2) = 23 − 2 = 21 YES (Step A)

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Now we have a number smaller than 7, and this number (4) is the remainder of dividing 186/7. So 186 minus 4, which is 182, must be a multiple of 7.

2057:. In other words, in 2 + 7 = 9, 7 is divisible by 7. So 2 and 9 must have the same remainder when divided by 7. The remainder is 2. 3614:

factors. For example, to determine divisibility by 36, check divisibility by 4 and by 9. Note that checking 3 and 12, or 2 and 18, would not be sufficient. A

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22,862,420 − (999,999 × 22) = 22,862,420 − 21,999,978 -> 862,420 + 22 = 862,442 862,442 -> 862 − 442 (Step B) = 420 -> 42 − (0×2) (Step A) = 42 YES

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Either of the last two rules may be used, depending on which is easier to perform. They correspond to the rule "subtract twice the last digit from the rest".

4407:{\displaystyle 10^{n}\equiv (-1)^{n}\equiv {\begin{cases}1,&{\mbox{if }}n{\mbox{ is even}}\\-1,&{\mbox{if }}n{\mbox{ is odd}}\end{cases}}{\pmod {11}}.} 2911: 367:

Subtract the quantity of the digits 2, 5, and 8 in the number from the quantity of the digits 1, 4, and 7 in the number. The result must be divisible by 3.

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Testing 871 this way reduces it to the divisibility of 91 using the second rule, and then 13 using that rule again, so we see that 871 is divisible by 13.

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Now, change 1 into the following digit in the sequence (3), add it to the second digit, and write the result instead of both: 3 + 1 =

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Remainder Test 13 (1, −3, −4, −1, 3, 4, cycle goes on.) If you are not comfortable with negative numbers, then use this sequence. (1, 10, 9, 12, 3, 4)

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This section will illustrate the basic method; all the rules can be derived following the same procedure. The following requires a basic grounding in

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Multiply the right-most digit by 5 and add to the rest of the numbers. If this sum is divisible by 7, then the original number is divisible by 7.

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that is just the same as converting 10×10 into a 3×10. And that is actually the same as subtracting 7×10 (clearly a multiple of 7) from 10×10.

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Alternatively, any number Q = 10c + d is divisible by n = 10a + b, such that gcd(n, 2, 5) = 1, if c + D(n)d = An for some integer A, where:

2232:

Is 355,341 divisible by seven? 355,341 − 341,341 = 14,000 (Step B) -> 014 − 000 (Step B) -> 14 = 1 − (4×2) (Step A) = 1 − 8 = −7 YES

1978:, that will indicate how much you should subtract from the original number to get a multiple of 7. In other words, you will find the 3950:

The piece wise form of D(n) and the sequence generated by it were first published by Bulgarian mathematician Ivan Stoykov in March 2020.

1864:, repeating with this sequence of multipliers as long as necessary (1, 3, 2, 6, 4, 5, 1, 3, 2, 6, 4, 5, ...), and adding the products (1× 939:

The sum of the ones digit, double the tens digit, four times the hundreds digit, and eight times the thousands digit is divisible by 16.

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Gardner, Martin (September 1962). "Mathematical Games: Tests that show whether a large number can be divided by a number from 2 to 12".

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15 is divisible by 3 at which point we can stop. Alternatively we can continue using the same method if the number is still too large:

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Subtracting 5 times the last digit from 2 times the rest of the number gives a multiple of 26. (Works because 52 is divisible by 26.)

2590: 1510: 5718: 5640: 5536: 2430:

Separate the number into digit pairs starting from the ones place. Prepend the number with 0 to complete the final pair if required.

2147:. Convert the divisor seven to the nines family by multiplying by seven. 7×7=49. Add one, drop the units digit and, take the 5, the 5655: 3421:{\displaystyle {\overline {a_{n}a_{n-1}...a_{1}a_{0}}}=\sum _{i=0}^{n}a_{i}10^{i}\equiv \sum _{i=0}^{n}(-1)^{i}a_{i}{\bmod {1}}1.} 2238:

Using quick alternating additions and subtractions: 42,341,530 -> 530 − 341 + 42 = 189 + 42 = 231 -> 23 − (1×2) = 21 YES

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If the number of digits is even, add the first and subtract the last digit from the rest. The result must be divisible by 11.

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Adding the digits of a number up, and then repeating the process with the result until only one digit remains, will give the

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16,499,205,854,376 → 1 + 6 + 4 + 9 + 9 + 2 + 0 + 5 + 8 + 5 + 4 + 3 + 7 + 6 sums to 69 → 6 + 9 = 15, which is divisible by 3.

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without performing the division, usually by examining its digits. Although there are divisibility tests for numbers in any

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7 − (1, 3, 2, −1, −3, −2, cycle repeats for the next six digits) Period: 6 digits. Recurring numbers: 1, 3, 2, −1, −3, −2

2174:

112 -> 11 − (2×2) = 11 − 4 = 7 YES 98 -> 9 − (8×2) = 9 − 16 = −7 YES 634 -> 63 − (4×2) = 63 − 8 = 55 NO

849:

Subtract 9 times the last digit from the rest. The result must be divisible by 13. (Works because 91 is divisible by 13).

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ends respectively in 1, 3, 7, or 9, then multiply by 9, 3, 7, or 1.) Then add 1 and divide by 10, denoting the result as

1921:. This second step may be skipped, except for the left most digit, but following it may facilitate calculations later on. 1772:

3 + 2 + 4 = 9 which is a multiple of 3. Therefore the original number is divisible by both 2 and 3 and is divisible by 6.

5603: 2203:

341,355 − 341,341 = 14 -> 1 − (4×2) = 1 − 8 = −7 YES 67,326 − 067,067 = 259 -> 25 − (9×2) = 25 − 18 = 7 YES

5906: 5806: 5896: 5702: 5668: 2022:

Repeat the procedure one more time: 1 becomes 3, which is added to the second digit (1): 3 + 1 =

783:

If the number of digits is odd, subtract the first and last digit from the rest. The result must be divisible by 11.

767:

Add 10 times the last digit to the rest. The result must be divisible by 11. (Works because 99 is divisible by 11).

2015:

Repeat the procedure, since the number is greater than 7. Now, 4 becomes 5, which must be added to 6. That is

1545:

492 ÷ 3 = 164 (If the number obtained by using the rule is divisible by 3, then the whole number is divisible by 3)

833:

Add 4 times the last digit to the rest. The result must be divisible by 13. (Works because 39 is divisible by 13).

441:

1,458: 1 + 4 + 5 + 8 = 18, so it is divisible by 3 and the last digit is even, hence the number is divisible by 6.

743:

Form the alternating sum of the digits, or equivalently sum(odd) - sum(even). The result must be divisible by 11.

572:

Adding 3 times the first digit to the next and then writing the rest gives a multiple of 7. (This works because 10

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which is also the difference between the sum of digits at odd positions and the sum of digits at even positions.

1654:), and seeing if it is either 0 or 5. If the last number is either 0 or 5, the entire number is divisible by 5. 86: 5375: 3049: 2268:(1, 3, 2, 6, 4, 5, cycle repeats for the next six digits) Period: 6 digits. Recurring numbers: 1, 3, 2, 6, 4, 5 1970:

or any recognizable multiple of 7, then the original number is a multiple of 7. If you obtain any number from

564:

Subtracting 9 times the last digit from the rest gives a multiple of 7. (Works because 91 is divisible by 7.)

556:

Subtracting 2 times the last digit from the rest gives a multiple of 7. (Works because 21 is divisible by 7.)

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2092 ÷ 4 = 523 (If the number that is obtained is divisible by 4, then the original number is divisible by 4)

705:

Subtracting 8 times the last digit from the rest gives a multiple of 9. (Works because 81 is divisible by 9)

375:

Subtracting 2 times the last digit from the rest gives a multiple of 3. (Works because 21 is divisible by 3)

3044:

In order to check divisibility by 11, consider the alternating sum of the digits. For example with 907,071:

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which is the rule "double the number formed by all but the last two digits, then add the last two digits".

4684: 4268:. For the higher powers of 10, they are congruent to 1 for even powers and congruent to −1 for odd powers: 3985: 3545: 2159:

Is 438,722,025 divisible by seven? Multiplier = 5. 4 3 8 7 2 2 0 2 5 42 37 46 37 6 40 37 27 YES

1170:

Suming 19 times the last digit to the rest gives a multiple of 21. (Works because 189 is divisible by 21).

2362:

Because the resulting 182 is less than six digits, we add zero's to the right side until it is six digits.

614:

Adding the last two digits to twice the rest gives a multiple of 7. (Works because 98 is divisible by 7.)

5750:, by Swami Sankaracarya, published by Motilal Banarsidass, Varanasi, India, 1965, Delhi, 1978. 367 pages. 4934: 4828: 912:

If the thousands digit is odd, the number formed by the last three digits must be 8 times an odd number.

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Divisibility properties of numbers can be determined in two ways, depending on the type of the divisor.

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If the hundreds digit is odd, the number obtained by the last two digits must be 4 times an odd number.

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For example, 7 does not divide 9 or 10, but does divide 98, which is close to 100. Thus, proceed from

3503: 682:

The sum of the ones digit, double the tens digit, and four times the hundreds digit is divisible by 8.

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Adding 5 times the last digit to the rest gives a multiple of 7. (Works because 49 is divisible by 7.)

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The number to be tested is 157514. First we separate the number into three digit pairs: 15, 75 and 14.

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The first few terms of the sequence, generated by D(n), are 1, 1, 5, 1, 10, 4, 12, 2, ... (sequence

1745:

Divisibility by 6 is determined by checking the original number to see if it is both an even number (

44: 4318: 3769: 3647:

in the table below. Likewise, since 10 × (28) = 280 = 1 mod 31 also, we obtain a complementary rule

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If the thousands digit is even, the number formed by the last three digits must be divisible by 16.

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The remainder of the sum when divided by 7 is the remainder of the given number when divided by 7.

2413:

The result −77 is divisible by seven, thus the original number 15751537186 is divisible by seven.

4984: 4737: 4557:{\displaystyle 1000\cdot a+100\cdot b+10\cdot c+1\cdot d\equiv (-1)a+(1)b+(-1)c+(1)d{\pmod {11}}} 2687: 2450: 28: 3610:

A number is divisible by a given divisor if it is divisible by the highest power of each of its

650:

If the hundreds digit is even, the number formed by the last two digits must be divisible by 8.

5328:

Since 10 × 5 ≡ 10 × (−2) ≡ 1 (mod 7) we can do the following:

3739:= 16×9 + 68 = 212, which is divisible by 53 (with quotient 4); so 689 is also divisible by 53. 3720:). The sum (or alternating sum) of the numbers have the same divisibility as the original one. 3615: 1406:

Subtract twice the last three digits from the rest. (Works because 2,001 is divisible by 29.)

1300:

Sum the digits in blocks of three from right to left. (Works because 999 is divisible by 27.)

1232:

Subtract twice the last three digits from the rest. (Works because 2,001 is divisible by 23.)

970:

Subtract the last two digits from two times the rest. (Works because 102 is divisible by 17.)

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The result −42 is divisible by seven, thus the original number 157514 is divisible by seven.

1641:

1720 ÷ 4 = 430 (If the result is divisible by 2, then the original number is divisible by 4)

1489:

376 ÷ 2 = 188 (If the last digit is divisible by 2, then the whole number is divisible by 2)

1324:

Subtract the last two digits from 8 times the rest. (Works because 108 is divisible by 27.)

498: 5761: 5461: 5031:; for divisibility other than by 2's and 5's the proofs rest on the basic fact that 10 mod 841:

Subtract the last two digits from four times the rest. The result must be divisible by 13.

778:

918,082: the number of digits is even (6) → 1808 + 9 − 2 = 1815: 81 + 1 − 5 = 77 = 7 × 11

8: 4091: 3968: 2249:

Is 634 divisible by seven? 634 -> 6×3 + 3 = 21 -> remainder 0 -> 0×3 + 4 = 4 NO

1134:

Subtracting twice the last digit from the rest gives a multiple of 21. (Works because (10

49: 35:, or base, and they are all different, this article presents rules and examples only for 5726: 3165: 2495:

The correctness of the method is then established by the following chain of equalities:

2426:

This is a non-recursive method to find the remainder left by a number on dividing by 7:

2068:/7 is 0), then adding (or subtracting) multiples of 7 cannot change that property. 1398:

Subtract 26 times the last digit from the rest. (Works because 261 is divisible by 29).

1224:

Subtract 16 times the last digit from the rest. (Works because 161 is divisible by 23).

978:

Add 2 times the last digit to 3 times the rest. Drop trailing zeroes. (Works because (10

786:

14,179: the number of digits is odd (5) → 417 − 1 − 9 = 407: 0 − 4 − 7 = −11 = −1 × 11.

751:

Add the digits in blocks of two from right to left. The result must be divisible by 11.

5799: 5497: 5493: 5028: 4025: 3635: 3145: 3125: 3025:{\displaystyle \sum _{k=1}^{n}(a_{2k}a_{2k-1}{\bmod {7}})\times (10^{2k-2}{\bmod {7}})} 72:

To test the divisibility of a number by a power of 2 or a power of 5 (2 or 5, in which

5871: 3963:

Many of the simpler rules can be produced using only algebraic manipulation, creating

1308:

Subtract 8 times the last digit from the rest. (Works because 81 is divisible by 27.)

954:

Subtract 5 times the last digit from the rest. (Works because 51 is divisible by 17.)

5851: 5829: 5810: 5698: 2200:

done by subtracting the first three digits from the last three digits. For example:

1390:

Add 9 times the last two digits to the rest. (Works because 899 is divisible by 29.)

1216:

Add 3 times the last two digits to the rest. (Works because 299 is divisible by 23.)

1088:

Add 4 times the last two digits to the rest. (Works because 399 is divisible by 19.)

5153:{\displaystyle 10^{n}=2^{n}\cdot 5^{n}\equiv 0{\pmod {2^{n}\mathrm {\ or\ } 5^{n}}}} 920:

Add the last two digits to four times the rest. The result must be divisible by 16.

5843: 5489: 5456: 3712:. If the number is too large, you can also break it down into several strings with 3622: 1650:

Divisibility by 5 is easily determined by checking the last digit in the number (47

1316:

Sum 19 times the last digit from the rest. (Works because 189 is divisible by 27).

360:

405 → 4 + 0 + 5 = 9 and 636 → 6 + 3 + 6 = 15 which both are clearly divisible by 3.

357:

The sum of the digits must be divisible by 3. (Works because 9 is divisible by 3).

16:

Shorthand way of determining whether a given number is divisible by a fixed divisor

5564:

A number is divisible by 2, 5 or 10 if and only if the number formed by the last

962:

Add 12 times the last digit to the rest. (Works because 119 is divisible by 17).

822: 806:

Subtract the last digit from twice the rest. The result must be divisible by 12.

635:

194,536: 19|45|36 ; (5x4) + (3x2) + (1x1) = 27, so it is not divisible by 7

626:

483,595: (4 × (−2)) + (8 × (−3)) + (3 × (−1)) + (5 × 2) + (9 × 3) + (5 × 1) = 7.

537: 5850:. Pure and Applied Undergraduate Texts. Vol. 3. American Mathematical Soc. 5759:

Dunkels, Andrejs, "Comments on note 82.53—a generalized test for divisibility",

4223:

Using 11 as an example, 11 divides 11 = 10 + 1. That means

1208:

Add 7 times the last digit to the rest. (Works because 69 is divisible by 23.)

872:

Add the last two digits to twice the rest. The result must be divisible by 14.

40: 5302:{\displaystyle x=10^{n}\cdot y+z\equiv z{\pmod {2^{n}\mathrm {\ or\ } 5^{n}}}} 4417:

Like the previous case, we can substitute powers of 10 with congruent values:

3676:

ends in 1, 3, 7, or 9, the following method can be used. Find any multiple of

3626:

This is not an issue for prime divisors because they have no smaller factors.

1781:

Divisibility by 7 can be tested by a recursive method. A number of the form 10

5890: 4202:{\displaystyle 100\cdot a+10\cdot b+1\cdot c\equiv (1)a+(1)b+(1)c{\pmod {3}}} 2229:

Is 634 divisible by seven? 634 -> 63 − (4×2) = 63 − 8 = 55 NO (Step A)

2226:

Is 98 divisible by seven? 98 -> 9 − (8×2) = 9 − 16 = −7 YES (Step A)

637:

204,540: 20|45|40 ; (6x4) + (3x2) + (5x1) = 35, so it is divisible by 7

83:

To test divisibility by any number expressed as the product of prime factors

5826:

Mathematical problems and proofs: combinatorics, number theory, and geometry

3982:

Using 3 as an example, 3 divides 9 = 10 − 1. That means

5875: 3723:

For example, to determine if 913 = 10×91 + 3 is divisible by 11, find that

3611: 1769:

Final digit 4 is even, so 324 is divisible by 2, and may be divisible by 6.

825:

of blocks of three from right to left. The result must be divisible by 13.

759:

Subtract the last digit from the rest. The result must be divisible by 11.

5674: 4628:

where in this case a is any integer, and b can range from 0 to 99. Next,

2900:{\displaystyle \sum _{k=1}^{n}(a_{2k}a_{2k-1}\times 10^{2k-2}){\bmod {7}}} 2136:= 0 + 15 + 0 + 6 = 21 (multiply and add). ANSWER: 1050 is divisible by 7. 147: 3979:

This method works for divisors that are factors of 10 − 1 = 9.

2246:

Is 98 divisible by seven? 98 -> 9 remainder 2 -> 2×3 + 8 = 14 YES

1924:

Now convert the first digit (3) into the following digit in the sequence

1699:

110 ÷ 5 = 22 (The result is the same as the original number divided by 5)

1415: 1348: 1333: 1294: 1271: 1256: 1241: 1202: 1187: 1128: 1097: 1038: 1023: 948: 898: 883: 858: 815: 792: 737: 714: 666:

Add the last digit to twice the rest. The result must be divisible by 8.

617:

483,595: 95 + (2 × 4835) = 9765: 65 + (2 × 97) = 259: 59 + (2 × 2) = 63.

5568:

digits is divisible by that number. See Richmond & Richmond (2009),

5501: 1736:

85 ÷ 5 = 17 (The result is the same as the original number divided by 5)

1518: 691: 644: 530: 432: 417: 408:

The sum of the ones digit and double the tens digit is divisible by 4.

384: 351: 336: 321: 1958:

begin with 0, which is useless. According to this, our 7 becomes

1603:

check that last digit is even, otherwise 6174 can't be divisible by 4.

1979: 1584:(Take the last two digits of the number, discarding any other digits) 1503: 5650: 5648: 62:

the divisibility is obvious; for others (such as examining the last

2433:

Calculate the remainders left by each digit pair on dividing by 7.

2113:

1050 → 105 − 0=105 → 10 − 10 = 0. ANSWER: 1050 is divisible by 7.

1513:, in which the divisor in question then becomes one less than the 5645: 2463:

The number 510,517,813 leaves a remainder of 1 on dividing by 7.

1542:

6 ÷ 3 = 2 (Check to see if the number received is divisible by 3)

36: 24: 4220:

This method works for divisors that are factors of 10 + 1 = 11.

2276:

Example: What is the remainder when 1036125837 is divided by 7?

2187:

111,111 / 7 = 15,873 222,222 / 7 = 31,746 999,999 / 7 = 142,857

2676:{\displaystyle {\overline {a_{2n}a_{2n-1}...a_{2}a_{1}}}\mod 7} 2440:

Calculate the remainders left by each product on dividing by 7.

1982:

of dividing the number by 7. For example, take the number

2184:

01 01 01 = 10,101 / 7 = 1,443 10 10 10 = 101,010 / 7 = 14,430

1820:

Another method is multiplication by 3. A number of the form 10

1718:(Take the last digit of the number, and check if it is 0 or 5) 1684:(Take the last digit of the number, and check if it is 0 or 5) 1354:

Add three times the last digit to the rest. (Works because (10

524:

1,458: (4 × 1) + (4 × 4) + (4 × 5) + 8 = 4 + 16 + 20 + 8 = 48

4731:

and after eliminating the known multiple of 7, the result is

3569: 3520: 3463:

Example: What is the remainder when 1234567 is divided by 13?

3406: 3213: 3010: 2972: 2888: 2779: 2460:

The number 194,536 leaves a remainder of 6 on dividing by 7.

2143:

Divisibility by seven can be tested by multiplication by the

1514: 398:

If the tens digit is even, the ones digit must be 0, 4, or 8.

32: 23:

is a shorthand and useful way of determining whether a given

5535:

This follows from Pascal's criterion. See Kisačanin (1998),

2053:

will necessarily produce the same remainder when divided by

1727:(If it is 5, take the remaining digits, discarding the last) 1693:(If it is 0, take the remaining digits, discarding the last) 1638:

860 ÷ 2 = 430 (Check to see if the result is divisible by 2)

1486:

6 ÷ 2 = 3 (Check to see if the last digit is divisible by 2)

540:

of blocks of three from right to left gives a multiple of 7

43:

explained and popularized these rules in his September 1962

4381: 3944: 3920: 1619:

7 + 2 = 9 (Add half of last digit to the penultimate digit)

5776: 3940: 2242:

Multiplication by 3 method of divisibility by 7, examples:

1587:

92 ÷ 4 = 23 (Check to see if the number is divisible by 4)

1017:

4,675: 467 × 3 + 5 × 2 = 1,411; 238: 23 × 3 + 8 × 2 = 85.

390:

The last two digits form a number that is divisible by 4.

76:

is a positive integer), one only need to look at the last

5694:

The Penguin dictionary of curious and interesting numbers

5531: 3442:

Example: What is the remainder when 321 is divided by 13?

2578:{\displaystyle {\overline {a_{2n}a_{2n-1}...a_{2}a_{1}}}} 2282:

Multiplication of the second rightmost digit = 3 × 3 = 9

400:

If the tens digit is odd, the ones digit must be 2 or 6.

5598: 5596: 5594: 5592: 5529: 5527: 5525: 5523: 5521: 5519: 5517: 5515: 5513: 5511: 1613:(Separate the last 2 digits from the rest of the number) 1044:

Add twice the last digit to the rest. (Works because (10

5748:

Vedic Mathematics: Sixteen Simple Mathematical Formulae

5641:

Section 3.4 (Divisibility Tests), Theorem 3.4.3, p. 107

5635: 5633: 5631: 4776:

Case where the last digit(s) is multiplied by a factor

4372: 4362: 4340: 4330: 3911: 3901: 3876: 3866: 3838: 3828: 3800: 3790: 3716:

digits each, satisfying either 10 = 1 or 10 = −1 (mod

3500:

A recursive method can be derived using the fact that

327:

No specific condition. Any integer is divisible by 1.

56: 5629: 5627: 5625: 5623: 5621: 5619: 5617: 5615: 5613: 5611: 5589: 5508: 5378: 5341: 5217: 5173: 5060: 4987: 4937: 4878: 4831: 4790: 4740: 4687: 4637: 4599: 4426: 4277: 4229: 4104: 4083:{\displaystyle 10^{n}\equiv 1^{n}\equiv 1{\pmod {3}}} 4034: 3988: 3748: 3548: 3506: 3243: 3199: 3168: 3148: 3128: 3052: 2914: 2805: 2690: 2593: 2504: 501: 452: 89: 3971:

each digit's power can be manipulated individually.

2033:

Note: The reason why this works is that if we have:

66:

digits) the result must be examined by other means.

2222:

Pohlman–Mass method of divisibility by 7, examples:

2196:This phenomenon forms the basis for Steps B and C. 1533:

4 + 9 + 2 = 15 (Add each individual digit together)

1103:It is divisible by 10, and the tens digit is even. 5798: 5608: 5424: 5362: 5301: 5201: 5152: 5008: 4967: 4917: 4858: 4811: 4761: 4720: 4667: 4617: 4556: 4406: 4260: 4201: 4082: 4016: 3967:and rearranging them. By writing a number as the 3929: 3583: 3534: 3420: 3227: 3177: 3154: 3134: 3109: 3024: 2899: 2790: 2675: 2577: 2279:Multiplication of the rightmost digit = 1 × 7 = 7 513: 487: 137: 5022: 998:; since 17 is a prime and 2 is coprime with 17, 3 677:34,152: Examine divisibility of just 152: 19 × 8 5888: 5842: 4216:Case where the alternating sum of digits is used 3663: 1635:1720 ÷ 2 = 860 (Divide the original number by 2) 1401:1,015: 101 - 5 x 26 = 101 - 130 = -29 = 29 x -1 1064:; since 19 is a prime and 2 is coprime with 19, 411:40,832: 2 × 3 + 2 = 8, which is divisible by 4. 5560: 5558: 5556: 5554: 5552: 5550: 5548: 5546: 5544: 2260:Finding remainder of a number when divided by 7 1319:1,026: 102 + 6 x 19 = 102 + 114 = 216 = 27 x 8 5579: 5577: 2489:10,000,000,000 mod 7 = 2^5 = 32; 32 mod 7 = 4 2417:Another digit pair method of divisibility by 7 1622:Since 9 isn't even, 6174 is not divisible by 4 1539:1 + 5 = 6 (Add each individual digit together) 1374:; the last number has the same remainder as 10 1154:; the last number has the same remainder as 10 1111:The last two digits are 00, 20, 40, 60 or 80. 931:The last four digits must be divisible by 16. 746:918,082: 9 − 1 + 8 − 0 + 8 − 2 = 22 = 2 × 11. 697:The sum of the digits must be divisible by 9. 592:; the last number has the same remainder as 10 5848:A Discrete Transition to Advanced Mathematics 4583:Case where only the last digit(s) are removed 4028:). The same for all the higher powers of 10: 1828:has the same remainder when divided by 7 as 3 1327:6,507: 65 × 8 − 7 = 520 − 7 = 513 = 27 × 19. 1288:1,248 : (124 ×2) - (8×5) = 208 = 26 × 8 1227:1,012: 101 - 2 x 16 = 101 - 32 = 69 = 23 x 3 5604:Section 3.4 (Divisibility Tests), p. 102–108 5541: 3958: 1211:3,128: 312 + 8 × 7 = 368. 36 + 8 × 7 = 92. 708:2,880: 288 - 0 x 8 = 288 - 0 = 288 = 9 x 32 5574: 2486:100,000,000 mod 7 = 2^4 = 16; 16 mod 7 = 2 2064:is a multiple of 7 (i.e.: the remainder of 1750: 770:627: 62 + 70 = 132: 13 + 20 = 33 = 3 × 11. 403:40,832: 3 is odd, and the last digit is 2. 342:The last digit is even (0, 2, 4, 6, or 8). 138:{\displaystyle p_{1}^{n}p_{2}^{m}p_{3}^{q}} 5425:{\displaystyle -2x\equiv y-2z{\pmod {7}},} 3110:{\displaystyle 9-0+7-0+7-1=22=2\times 11,} 2140:Vedic method of divisibility by osculation 1262:The last two digits are 00, 25, 50 or 75. 674:The last three digits are divisible by 8. 5823: 2669: 2668: 1917:, respectively. In this example, we get: 1493: 1432:270: it is divisible by 2, by 3 and by 5 1173:441: 44 + 1 x 19 = 44 + 19 = 63 = 21 x 3 965:221: 22 + 1 x 12 = 22 + 12 = 34 = 17 x 2 543:1,369,851: 851 − 369 + 1 = 483 = 7 × 69 4571:Case where only the last digit(s) matter 4212:which is exactly the sum of the digits. 2449: 2163:Pohlman–Mass method of divisibility by 7 1454: 1106:360: is divisible by 10, and 6 is even. 942:157,648: 7 × 8 + 6 × 4 + 4 × 2 + 8 = 96 378:405: 40 - 5 x 2 = 40 - 10 = 30 = 3 x 10 5796: 5774: 5710: 5479: 4918:{\displaystyle (21-1)\cdot a+2\cdot b.} 4261:{\displaystyle 10\equiv -1{\pmod {11}}} 3228:{\displaystyle 10\equiv -1{\bmod {1}}1} 2483:1,000,000 mod 7 = 2^3 = 8; 8 mod 7 = 1 2385:The number to be tested is 15751537186. 1966:If through this procedure you obtain a 836:637: 63 + 7 × 4 = 91, 9 + 1 × 4 = 13. 5889: 5801:Introduction to analytic number theory 2319:Digit pair method of divisibility by 7 1746: 1696:11 × 2 = 22 (Multiply the result by 2) 700:2,880: 2 + 8 + 8 + 0 = 18: 1 + 8 = 9. 5690: 4822:after multiplying by 2, this becomes 4721:{\displaystyle 98\cdot a+2\cdot a+b,} 4017:{\displaystyle 10\equiv 1{\pmod {3}}} 3969:sum of each digit times a power of 10 3634:The goal is to find an inverse to 10 3605: 3584:{\displaystyle 10^{-1}=4{\bmod {1}}3} 3430: 3034: 2288:Fourth rightmost digit = 5 × −1 = −5 1730:8 × 2 = 16 (Multiply the result by 2) 1424:270: it is divisible by 3 and by 10. 1280:156: it is divisible by 2 and by 13. 1196:352: it is divisible by 2 and by 11. 5768: 2306:Tenth rightmost digit = 1 × −1 = −1 2297:Seventh rightmost digit = 6 × 1 = 6 2294:Sixth rightmost digit = 1 × −2 = −2 2291:Fifth rightmost digit = 2 × −3 = −6 1883:The simplification goes as follows: 1776: 1740: 1645: 1549: 1459: 1440:270: it is divisible by 2 and by 15 1429:It is divisible by 2, by 3 and by 5 1342:140: it is divisible by 4 and by 7. 1250:552: it is divisible by 3 and by 8. 1181:231: it is divisible by 3 and by 7. 1072:is divisible by 19 if and only if 10 1032:342: it is divisible by 2 and by 9. 1006:is divisible by 17 if and only if 10 892:390: it is divisible by 3 and by 5. 867:224: it is divisible by 2 and by 7. 801:324: it is divisible by 3 and by 4. 5666: 5411: 5260: 5111: 4968:{\displaystyle -1\cdot a+2\cdot b,} 4859:{\displaystyle 20\cdot a+2\cdot b,} 4546: 4393: 4250: 4191: 4072: 4006: 2364:Then we apply our algorithm again: 2300:Eighth rightmost digit = 3 × 3 = 9 2285:Third rightmost digit = 8 × 2 = 16 1616:4 ÷ 2 = 2 (last digit divided by 2) 1448:270: it is divisible by 5 and by 6 1393:5,510: 55 + 10 × 9 = 145 = 5 × 29. 731:130: it is divisible by 2 and by 5 488:{\displaystyle 10^{n}=4{\pmod {6}}} 477: 57:Divisibility rules for numbers 1–30 13: 5807:Undergraduate Texts in Mathematics 5494:10.1038/scientificamerican0962-232 5278: 5275: 5129: 5126: 3189:Proof of correctness of the method 2467:Proof of correctness of the method 2041:is a multiple of any given number 1409:2,086,956: 2,086 − 956 × 2 = 174. 1235:2,068,965: 2,068 − 965 × 2 = 138. 14: 5918: 5865: 5439:is divisible by 7 if and only if 3629: 3535:{\displaystyle 10=-3{\bmod {1}}3} 1809:is divisible by 7 if and only if 1801:Similarly a number of the form 10 1789:is divisible by 7 if and only if 1733:16 + 1 = 17 (Add 1 to the result) 1413: 1346: 1303:2,644,272: 2 + 644 + 272 = 918. 1292: 1269: 1200: 1126: 1122:480: it is divisible by 4 and 5. 1095: 1036: 946: 896: 856: 813: 790: 735: 712: 5809:. Vol. 1. Springer-Verlag. 5639:Richmond & Richmond (2009), 5602:Richmond & Richmond (2009), 5202:{\displaystyle 10^{n}\cdot y+z,} 3975:Case where all digits are summed 3235:, we can write for any integer: 1840:them successively by the digits 1421:It is divisible by 3 and by 10. 1277:It is divisible by 2 and by 13. 1265:134,250: 50 is divisible by 25. 1193:It is divisible by 2 and by 11. 828:2,911,272: 272 − 911 + 2 = −637 689: 642: 567:483: 48 − (3 × 9) = 21 = 7 × 3. 559:483: 48 − (3 × 2) = 42 = 7 × 6. 551:483: 48 + (3 × 5) = 63 = 7 × 9. 528: 430: 382: 349: 5753: 5740: 5404: 5253: 5104: 4668:{\displaystyle (98+2)\cdot a+b} 4539: 4386: 4243: 4184: 4065: 3999: 3162:since multiplying the whole by 3119:so 907,071 is divisible by 11. 2664: 1509:This can be generalized to any 1437:It is divisible by 2 and by 15 1339:It is divisible by 4 and by 7. 1247:It is divisible by 3 and by 8. 1178:It is divisible by 3 and by 7. 1029:It is divisible by 2 and by 9. 889:It is divisible by 3 and by 5. 864:It is divisible by 2 and by 7. 809:324: 32 × 2 − 4 = 60 = 5 × 12. 798:It is divisible by 3 and by 4. 685:34,152: 4 × 1 + 5 × 2 + 2 = 16 470: 438:It is divisible by 2 and by 3. 5716: 5684: 5660: 5473: 5415: 5405: 5295: 5254: 5146: 5105: 5023:Proof using modular arithmetic 4891: 4879: 4650: 4638: 4550: 4540: 4532: 4526: 4517: 4508: 4499: 4493: 4484: 4475: 4397: 4387: 4301: 4291: 4254: 4244: 4195: 4185: 4177: 4171: 4162: 4156: 4147: 4141: 4076: 4066: 4010: 4000: 3758: 3752: 3386: 3376: 3019: 2987: 2981: 2936: 2884: 2827: 2775: 2750: 2715: 2691: 1990:First, change the 8 into a 1: 1445:It is divisible by 5 and by 6 728:It is divisible by 2 and by 5 481: 471: 393:40,832: 32 is divisible by 4. 1: 5883:Divisibility rules for 2–100. 5824:Kisačanin, Branislav (1998). 5467: 3664:Generalized divisibility rule 2348:Then we apply the algorithm: 1887:Take for instance the number 5775:Stoykov, Ivan (March 2020). 5363:{\displaystyle 10\cdot y+z,} 4978:and multiplying by −1 gives 4812:{\displaystyle 10\cdot a+b,} 4618:{\displaystyle 100\cdot a+b} 3668:To test for divisibility by 3597: 3303: 2659: 2570: 1950:6 + 1 = 1119:It is divisible by 4 and 5. 7: 5846:; Richmond, Thomas (2009). 5450: 5051:digits need to be checked. 5009:{\displaystyle a-2\cdot b.} 4762:{\displaystyle 2\cdot a+b,} 3465:Using the second sequence, 3460:Remainder = −17 mod 13 = 9 2791:{\displaystyle {\bmod {7}}} 2266:Minimum magnitude sequence 934:157,648: 7,648 = 478 × 16. 762:627: 62 − 7 = 55 = 5 × 11. 754:627: 6 + 27 = 33 = 3 × 11. 45:"Mathematical Games" column 10: 5923: 5881:Stupid Divisibility Tricks 5790: 3444:Using the first sequence, 3185:does not change anything. 2498:Let N be the given number 2303:Ninth rightmost digit = 0 2120:1050 → 0501 (reverse) → 0× 1928:In our example, 3 becomes 1893:Change all occurrences of 1632:1720 (The original number) 1600:6174 (the original number) 1575:2092 (The original number) 1511:standard positional system 1091:6,935: 69 + 35 × 4 = 209. 925:1,168: 11 × 4 + 68 = 112. 723:130: the ones digit is 0. 426:495: the last digit is 5. 423:The last digit is 0 or 5. 5907:Mathematics-related lists 5719:""Divisibility by Seven" 4928:Eliminating the 21 gives 3959:Proof using basic algebra 3953: 3735:= (53×3+1)÷10 = 16. Then 3727:= (11×9+1)÷10 = 10. Then 3660:the rest of the number). 3122:We can either start with 1766:324 (The original number) 1675:110 (The original number) 1662:divided by 5 (125/5=25). 1530:492 (The original number) 1474:376 (The original number) 1219:1,725: 17 + 25 × 3 = 92. 973:4,675: 46 × 2 − 75 = 17. 877:1,764: 17 × 2 + 64 = 98. 5897:Elementary number theory 5797:Apostol, Tom M. (1976). 5312:and the divisibility of 5035:is invertible if 10 and 2411:× 60) = −180 + 103 = −77 1709:85 (The original number) 39:, or base 10, numbers. 27:is divisible by a fixed 5316:is the same as that of 2480:10,000 mod 7 = 2^2 = 4 2101:50×10 − 1×10= 2094:40×10 − 5×10= 2087:60×10 − 4×10= 2080:20×10 − 6×10= 2060:Therefore, if a number 957:221: 22 − 1 × 5 = 17. 312:Divisibility condition 296: 291: 286: 281: 276: 271: 266: 261: 256: 251: 246: 241: 236: 231: 226: 221: 216: 211: 206: 201: 196: 80:digits of that number. 5902:Division (mathematics) 5765:84, March 2000, 79-81. 5426: 5364: 5303: 5203: 5154: 5039:are relatively prime. 5010: 4969: 4919: 4860: 4813: 4763: 4722: 4669: 4619: 4558: 4408: 4262: 4203: 4084: 4018: 3931: 3616:table of prime factors 3585: 3536: 3422: 3375: 3331: 3229: 3179: 3156: 3136: 3111: 3026: 2935: 2901: 2826: 2792: 2714: 2677: 2579: 2454: 1704:If the last digit is 5 1670:If the last digit is 0 1494:Divisibility by 3 or 9 1385:348: 34 + 8 × 3 = 58. 1311:621: 62 − 1 × 8 = 54. 1083:437: 43 + 7 × 2 = 57. 923:176: 1 × 4 + 76 = 80. 852:637: 63 - 7 × 9 = 0. 844:923: 9 × 4 − 23 = 13. 669:56: (5 × 2) + 6 = 16. 515: 514:{\displaystyle n>1} 489: 191: 186: 181: 176: 171: 166: 161: 156: 151: 139: 5872:Divisibility Criteria 5691:Wells, David (1997), 5427: 5365: 5304: 5204: 5155: 5011: 4970: 4920: 4861: 4814: 4764: 4723: 4670: 4620: 4559: 4409: 4263: 4204: 4085: 4019: 3932: 3586: 3537: 3423: 3355: 3311: 3230: 3180: 3157: 3137: 3112: 3027: 2915: 2902: 2806: 2793: 2694: 2678: 2580: 2453: 2443:Add these remainders. 2437:again by 1 and so on. 2190:576,576 / 7 = 82,368 2155:Vedic method example: 2117:Second method example 1483:(Take the last digit) 1455:Step-by-step examples 1331: 1254: 1239: 1185: 1165:168: 16 − 8 × 2 = 0. 1021: 881: 875:364: 3 × 2 + 64 = 70. 720:The last digit is 0. 603:483: 4 × 3 + 8 = 20, 516: 490: 330:2 is divisible by 1. 140: 5762:Mathematical Gazette 5462:Parity (mathematics) 5376: 5339: 5215: 5171: 5058: 4985: 4935: 4876: 4829: 4788: 4738: 4685: 4678:and again expanding 4635: 4597: 4424: 4275: 4227: 4102: 4032: 3986: 3746: 3546: 3504: 3495:× 1 = 178 mod 13 = 9 3241: 3197: 3166: 3146: 3126: 3050: 2912: 2803: 2688: 2591: 2502: 2110:First method example 915:3408: 408 = 8 x 51. 608:63: 6 × 3 + 3 = 21. 605:203: 2 × 3 + 0 = 6, 499: 450: 415: 334: 319: 87: 5721:Mudd Math Fun Facts 5482:Scientific American 5447:is divisible by 7. 305: 134: 119: 104: 50:Scientific American 5670:Divisibility Tests 5654:Kisačanin (1998), 5422: 5360: 5299: 5199: 5150: 5029:modular arithmetic 5006: 4965: 4915: 4856: 4809: 4759: 4718: 4665: 4615: 4554: 4404: 4380: 4376: 4366: 4344: 4334: 4258: 4199: 4080: 4026:modular arithmetic 4014: 3927: 3919: 3915: 3905: 3880: 3870: 3842: 3832: 3804: 3794: 3606:Composite divisors 3581: 3532: 3431:Divisibility by 13 3418: 3225: 3178:{\displaystyle -1} 3175: 3152: 3132: 3107: 3035:Divisibility by 11 3022: 2897: 2788: 2673: 2575: 2455: 2270:Positive sequence 661:352: 52 = 4 x 13. 511: 485: 345:1,294: 4 is even. 304: 135: 120: 105: 90: 5857:978-0-8218-4789-3 5844:Richmond, Bettina 5835:978-0-306-45967-2 5816:978-0-387-90163-3 5667:Loy, Jim (1999), 5283: 5274: 5134: 5125: 4375: 4365: 4343: 4333: 3926: 3914: 3904: 3879: 3869: 3841: 3831: 3803: 3793: 3680:ending in 9. (If 3306: 3193:Considering that 3155:{\displaystyle +} 3135:{\displaystyle -} 2662: 2573: 2323:This method uses 2312:33 modulus 7 = 5 1777:Divisibility by 7 1741:Divisibility by 6 1646:Divisibility by 5 1550:Divisibility by 4 1460:Divisibility by 2 1452: 1451: 21:divisibility rule 5914: 5861: 5839: 5828:. Plenum Press. 5820: 5804: 5785: 5784: 5772: 5766: 5757: 5751: 5744: 5738: 5737: 5735: 5734: 5725:. Archived from 5714: 5708: 5707: 5688: 5682: 5681: 5673:, archived from 5664: 5658: 5652: 5643: 5637: 5606: 5600: 5587: 5583:Apostol (1976), 5581: 5572: 5562: 5539: 5533: 5506: 5505: 5477: 5457:Division by zero 5431: 5429: 5428: 5423: 5418: 5369: 5367: 5366: 5361: 5308: 5306: 5305: 5300: 5298: 5294: 5293: 5284: 5281: 5272: 5270: 5269: 5233: 5232: 5208: 5206: 5205: 5200: 5183: 5182: 5159: 5157: 5156: 5151: 5149: 5145: 5144: 5135: 5132: 5123: 5121: 5120: 5096: 5095: 5083: 5082: 5070: 5069: 5015: 5013: 5012: 5007: 4974: 4972: 4971: 4966: 4924: 4922: 4921: 4916: 4865: 4863: 4862: 4857: 4818: 4816: 4815: 4810: 4768: 4766: 4765: 4760: 4727: 4725: 4724: 4719: 4674: 4672: 4671: 4666: 4624: 4622: 4621: 4616: 4563: 4561: 4560: 4555: 4553: 4413: 4411: 4410: 4405: 4400: 4384: 4383: 4377: 4373: 4367: 4363: 4345: 4341: 4335: 4331: 4309: 4308: 4287: 4286: 4267: 4265: 4264: 4259: 4257: 4208: 4206: 4205: 4200: 4198: 4089: 4087: 4086: 4081: 4079: 4057: 4056: 4044: 4043: 4023: 4021: 4020: 4015: 4013: 3936: 3934: 3933: 3928: 3924: 3923: 3922: 3916: 3912: 3906: 3902: 3881: 3877: 3871: 3867: 3843: 3839: 3833: 3829: 3805: 3801: 3795: 3791: 3708:is divisible by 3700:is divisible by 3688:. Then a number 3590: 3588: 3587: 3582: 3577: 3576: 3561: 3560: 3541: 3539: 3538: 3533: 3528: 3527: 3427: 3425: 3424: 3419: 3414: 3413: 3404: 3403: 3394: 3393: 3374: 3369: 3351: 3350: 3341: 3340: 3330: 3325: 3307: 3302: 3301: 3300: 3291: 3290: 3272: 3271: 3256: 3255: 3245: 3234: 3232: 3231: 3226: 3221: 3220: 3184: 3182: 3181: 3176: 3161: 3159: 3158: 3153: 3141: 3139: 3138: 3133: 3116: 3114: 3113: 3108: 3031: 3029: 3028: 3023: 3018: 3017: 3008: 3007: 2980: 2979: 2970: 2969: 2951: 2950: 2934: 2929: 2906: 2904: 2903: 2898: 2896: 2895: 2883: 2882: 2861: 2860: 2842: 2841: 2825: 2820: 2797: 2795: 2794: 2789: 2787: 2786: 2774: 2773: 2749: 2748: 2730: 2729: 2713: 2708: 2682: 2680: 2679: 2674: 2663: 2658: 2657: 2656: 2647: 2646: 2628: 2627: 2609: 2608: 2595: 2584: 2582: 2581: 2576: 2574: 2569: 2568: 2567: 2558: 2557: 2539: 2538: 2520: 2519: 2506: 520: 518: 517: 512: 494: 492: 491: 486: 484: 462: 461: 306: 303: 144: 142: 141: 136: 133: 128: 118: 113: 103: 98: 5922: 5921: 5917: 5916: 5915: 5913: 5912: 5911: 5887: 5886: 5868: 5858: 5836: 5817: 5793: 5788: 5773: 5769: 5758: 5754: 5745: 5741: 5732: 5730: 5717:Su, Francis E. 5715: 5711: 5705: 5689: 5685: 5677:on 2007-10-10, 5665: 5661: 5653: 5646: 5638: 5609: 5601: 5590: 5582: 5575: 5563: 5542: 5534: 5509: 5478: 5474: 5470: 5453: 5403: 5377: 5374: 5373: 5340: 5337: 5336: 5289: 5285: 5271: 5265: 5261: 5252: 5228: 5224: 5216: 5213: 5212: 5178: 5174: 5172: 5169: 5168: 5140: 5136: 5122: 5116: 5112: 5103: 5091: 5087: 5078: 5074: 5065: 5061: 5059: 5056: 5055: 5025: 4986: 4983: 4982: 4936: 4933: 4932: 4877: 4874: 4873: 4830: 4827: 4826: 4789: 4786: 4785: 4739: 4736: 4735: 4686: 4683: 4682: 4636: 4633: 4632: 4598: 4595: 4594: 4538: 4425: 4422: 4421: 4385: 4379: 4378: 4371: 4361: 4359: 4347: 4346: 4339: 4329: 4327: 4314: 4313: 4304: 4300: 4282: 4278: 4276: 4273: 4272: 4242: 4228: 4225: 4224: 4183: 4103: 4100: 4099: 4064: 4052: 4048: 4039: 4035: 4033: 4030: 4029: 3998: 3987: 3984: 3983: 3961: 3956: 3918: 3917: 3910: 3900: 3898: 3883: 3882: 3875: 3865: 3863: 3845: 3844: 3837: 3827: 3825: 3807: 3806: 3799: 3789: 3787: 3765: 3764: 3747: 3744: 3743: 3704:if and only if 3666: 3651: + 28 3639:rule for using 3632: 3618:may be useful. 3608: 3600: 3572: 3568: 3553: 3549: 3547: 3544: 3543: 3523: 3519: 3505: 3502: 3501: 3496: 3466: 3464: 3459: 3445: 3443: 3433: 3409: 3405: 3399: 3395: 3389: 3385: 3370: 3359: 3346: 3342: 3336: 3332: 3326: 3315: 3296: 3292: 3286: 3282: 3261: 3257: 3251: 3247: 3246: 3244: 3242: 3239: 3238: 3216: 3212: 3198: 3195: 3194: 3167: 3164: 3163: 3147: 3144: 3143: 3127: 3124: 3123: 3051: 3048: 3047: 3037: 3013: 3009: 2994: 2990: 2975: 2971: 2956: 2952: 2943: 2939: 2930: 2919: 2913: 2910: 2909: 2891: 2887: 2869: 2865: 2847: 2843: 2834: 2830: 2821: 2810: 2804: 2801: 2800: 2782: 2778: 2760: 2756: 2735: 2731: 2722: 2718: 2709: 2698: 2689: 2686: 2685: 2652: 2648: 2642: 2638: 2614: 2610: 2601: 2597: 2596: 2594: 2592: 2589: 2588: 2563: 2559: 2553: 2549: 2525: 2521: 2512: 2508: 2507: 2505: 2503: 2500: 2499: 2412: 2386: 2384: 2377: 2363: 2361: 2347: 2345: 2335:pattern on the 2314: 2313: 2311: 2310: 2308: 2307: 2305: 2304: 2302: 2301: 2299: 2298: 2296: 2295: 2293: 2292: 2290: 2289: 2287: 2286: 2284: 2283: 2281: 2280: 2278: 2277: 2275: 2269: 2267: 2265: 2257: 2253: 2250: 2247: 2239: 2236: 2233: 2230: 2227: 2215: 2204: 2191: 2188: 2185: 2182: 2175: 2165: 2160: 2142: 2119: 2112: 1872: + 3× 1868: + 7× 1779: 1743: 1668: 1648: 1568: 1552: 1496: 1462: 1457: 876: 823:alternating sum 538:alternating sum 500: 497: 496: 469: 457: 453: 451: 448: 447: 399: 361: 302: 301: 129: 124: 114: 109: 99: 94: 88: 85: 84: 59: 17: 12: 11: 5: 5920: 5910: 5909: 5904: 5899: 5885: 5884: 5878: 5867: 5866:External links 5864: 5863: 5862: 5856: 5840: 5834: 5821: 5815: 5792: 5789: 5787: 5786: 5777:"OEIS A333448" 5767: 5752: 5739: 5709: 5703: 5697:, p. 51, 5683: 5659: 5644: 5607: 5588: 5573: 5540: 5507: 5488:(3): 232–246. 5471: 5469: 5466: 5465: 5464: 5459: 5452: 5449: 5433: 5432: 5421: 5417: 5414: 5410: 5407: 5402: 5399: 5396: 5393: 5390: 5387: 5384: 5381: 5359: 5356: 5353: 5350: 5347: 5344: 5310: 5309: 5297: 5292: 5288: 5280: 5277: 5268: 5264: 5259: 5256: 5251: 5248: 5245: 5242: 5239: 5236: 5231: 5227: 5223: 5220: 5198: 5195: 5192: 5189: 5186: 5181: 5177: 5161: 5160: 5148: 5143: 5139: 5131: 5128: 5119: 5115: 5110: 5107: 5102: 5099: 5094: 5090: 5086: 5081: 5077: 5073: 5068: 5064: 5047:Only the last 5024: 5021: 5017: 5016: 5005: 5002: 4999: 4996: 4993: 4990: 4976: 4975: 4964: 4961: 4958: 4955: 4952: 4949: 4946: 4943: 4940: 4926: 4925: 4914: 4911: 4908: 4905: 4902: 4899: 4896: 4893: 4890: 4887: 4884: 4881: 4867: 4866: 4855: 4852: 4849: 4846: 4843: 4840: 4837: 4834: 4820: 4819: 4808: 4805: 4802: 4799: 4796: 4793: 4770: 4769: 4758: 4755: 4752: 4749: 4746: 4743: 4729: 4728: 4717: 4714: 4711: 4708: 4705: 4702: 4699: 4696: 4693: 4690: 4676: 4675: 4664: 4661: 4658: 4655: 4652: 4649: 4646: 4643: 4640: 4626: 4625: 4614: 4611: 4608: 4605: 4602: 4565: 4564: 4552: 4549: 4545: 4542: 4537: 4534: 4531: 4528: 4525: 4522: 4519: 4516: 4513: 4510: 4507: 4504: 4501: 4498: 4495: 4492: 4489: 4486: 4483: 4480: 4477: 4474: 4471: 4468: 4465: 4462: 4459: 4456: 4453: 4450: 4447: 4444: 4441: 4438: 4435: 4432: 4429: 4415: 4414: 4403: 4399: 4396: 4392: 4389: 4382: 4370: 4360: 4358: 4355: 4352: 4349: 4348: 4338: 4328: 4326: 4323: 4320: 4319: 4317: 4312: 4307: 4303: 4299: 4296: 4293: 4290: 4285: 4281: 4256: 4253: 4249: 4246: 4241: 4238: 4235: 4232: 4210: 4209: 4197: 4194: 4190: 4187: 4182: 4179: 4176: 4173: 4170: 4167: 4164: 4161: 4158: 4155: 4152: 4149: 4146: 4143: 4140: 4137: 4134: 4131: 4128: 4125: 4122: 4119: 4116: 4113: 4110: 4107: 4078: 4075: 4071: 4068: 4063: 4060: 4055: 4051: 4047: 4042: 4038: 4012: 4009: 4005: 4002: 3997: 3994: 3991: 3960: 3957: 3955: 3952: 3921: 3909: 3899: 3897: 3894: 3891: 3888: 3885: 3884: 3874: 3864: 3862: 3859: 3856: 3853: 3850: 3847: 3846: 3836: 3826: 3824: 3821: 3818: 3815: 3812: 3809: 3808: 3798: 3788: 3786: 3783: 3780: 3777: 3774: 3771: 3770: 3768: 3763: 3760: 3757: 3754: 3751: 3665: 3662: 3643: − 3 3631: 3630:Prime divisors 3628: 3607: 3604: 3599: 3596: 3580: 3575: 3571: 3567: 3564: 3559: 3556: 3552: 3531: 3526: 3522: 3518: 3515: 3512: 3509: 3497:Remainder = 9 3432: 3429: 3417: 3412: 3408: 3402: 3398: 3392: 3388: 3384: 3381: 3378: 3373: 3368: 3365: 3362: 3358: 3354: 3349: 3345: 3339: 3335: 3329: 3324: 3321: 3318: 3314: 3310: 3305: 3299: 3295: 3289: 3285: 3281: 3278: 3275: 3270: 3267: 3264: 3260: 3254: 3250: 3224: 3219: 3215: 3211: 3208: 3205: 3202: 3174: 3171: 3151: 3131: 3106: 3103: 3100: 3097: 3094: 3091: 3088: 3085: 3082: 3079: 3076: 3073: 3070: 3067: 3064: 3061: 3058: 3055: 3036: 3033: 3021: 3016: 3012: 3006: 3003: 3000: 2997: 2993: 2989: 2986: 2983: 2978: 2974: 2968: 2965: 2962: 2959: 2955: 2949: 2946: 2942: 2938: 2933: 2928: 2925: 2922: 2918: 2894: 2890: 2886: 2881: 2878: 2875: 2872: 2868: 2864: 2859: 2856: 2853: 2850: 2846: 2840: 2837: 2833: 2829: 2824: 2819: 2816: 2813: 2809: 2785: 2781: 2777: 2772: 2769: 2766: 2763: 2759: 2755: 2752: 2747: 2744: 2741: 2738: 2734: 2728: 2725: 2721: 2717: 2712: 2707: 2704: 2701: 2697: 2693: 2672: 2667: 2661: 2655: 2651: 2645: 2641: 2637: 2634: 2631: 2626: 2623: 2620: 2617: 2613: 2607: 2604: 2600: 2572: 2566: 2562: 2556: 2552: 2548: 2545: 2542: 2537: 2534: 2531: 2528: 2524: 2518: 2515: 2511: 2477:100 mod 7 = 2 2448: 2447: 2444: 2441: 2438: 2434: 2431: 2315:Remainder = 5 2254: 2251: 2248: 2245: 2237: 2234: 2231: 2228: 2225: 2213: 2202: 2189: 2186: 2183: 2180: 2173: 2158: 2107: 2106: 2099: 2092: 2085: 2028: 2027: 2020: 2013: 2005:6 becomes now 1995: 1964: 1963: 1955: 1947: 1933: 1922: 1891: 1813: + 5 1793: − 2 1778: 1775: 1774: 1773: 1770: 1767: 1751:divisible by 3 1747:divisible by 2 1742: 1739: 1738: 1737: 1734: 1731: 1728: 1719: 1710: 1701: 1700: 1697: 1694: 1685: 1676: 1647: 1644: 1643: 1642: 1639: 1636: 1633: 1624: 1623: 1620: 1617: 1614: 1604: 1601: 1592: 1591: 1588: 1585: 1576: 1551: 1548: 1547: 1546: 1543: 1540: 1537: 1534: 1531: 1495: 1492: 1491: 1490: 1487: 1484: 1475: 1461: 1458: 1456: 1453: 1450: 1449: 1446: 1442: 1441: 1438: 1434: 1433: 1430: 1426: 1425: 1422: 1419: 1411: 1410: 1407: 1403: 1402: 1399: 1395: 1394: 1391: 1387: 1386: 1383: 1352: 1344: 1343: 1340: 1337: 1329: 1328: 1325: 1321: 1320: 1317: 1313: 1312: 1309: 1305: 1304: 1301: 1298: 1290: 1289: 1286: 1282: 1281: 1278: 1275: 1267: 1266: 1263: 1260: 1252: 1251: 1248: 1245: 1237: 1236: 1233: 1229: 1228: 1225: 1221: 1220: 1217: 1213: 1212: 1209: 1206: 1198: 1197: 1194: 1191: 1183: 1182: 1179: 1175: 1174: 1171: 1167: 1166: 1163: 1132: 1124: 1123: 1120: 1116: 1115: 1112: 1108: 1107: 1104: 1101: 1093: 1092: 1089: 1085: 1084: 1081: 1042: 1034: 1033: 1030: 1027: 1019: 1018: 1015: 975: 974: 971: 967: 966: 963: 959: 958: 955: 952: 944: 943: 940: 936: 935: 932: 928: 927: 921: 917: 916: 913: 909: 908: 907:254,176: 176. 905: 902: 894: 893: 890: 887: 879: 878: 873: 869: 868: 865: 862: 854: 853: 850: 846: 845: 842: 838: 837: 834: 830: 829: 826: 819: 811: 810: 807: 803: 802: 799: 796: 788: 787: 784: 780: 779: 776: 772: 771: 768: 764: 763: 760: 756: 755: 752: 748: 747: 744: 741: 733: 732: 729: 725: 724: 721: 718: 710: 709: 706: 702: 701: 698: 695: 687: 686: 683: 679: 678: 675: 671: 670: 667: 663: 662: 659: 655: 654: 651: 648: 640: 639: 633: 628: 627: 624: 619: 618: 615: 611: 610: 601: 569: 568: 565: 561: 560: 557: 553: 552: 549: 545: 544: 541: 534: 526: 525: 522: 510: 507: 504: 483: 480: 476: 473: 468: 465: 460: 456: 443: 442: 439: 436: 428: 427: 424: 421: 413: 412: 409: 405: 404: 401: 395: 394: 391: 388: 380: 379: 376: 372: 371: 368: 364: 363: 358: 355: 347: 346: 343: 340: 332: 331: 328: 325: 317: 316: 313: 310: 300: 299: 294: 289: 284: 279: 274: 269: 264: 259: 254: 249: 244: 239: 234: 229: 224: 219: 214: 209: 204: 199: 194: 189: 184: 179: 174: 169: 164: 159: 154: 148: 132: 127: 123: 117: 112: 108: 102: 97: 93: 58: 55: 41:Martin Gardner 15: 9: 6: 4: 3: 2: 5919: 5908: 5905: 5903: 5900: 5898: 5895: 5894: 5892: 5882: 5879: 5877: 5873: 5870: 5869: 5859: 5853: 5849: 5845: 5841: 5837: 5831: 5827: 5822: 5818: 5812: 5808: 5803: 5802: 5795: 5794: 5782: 5778: 5771: 5764: 5763: 5756: 5749: 5743: 5729:on 2019-06-13 5728: 5724: 5722: 5713: 5706: 5704:9780140261493 5700: 5696: 5695: 5687: 5680: 5676: 5672: 5671: 5663: 5657: 5651: 5649: 5642: 5636: 5634: 5632: 5630: 5628: 5626: 5624: 5622: 5620: 5618: 5616: 5614: 5612: 5605: 5599: 5597: 5595: 5593: 5586: 5580: 5578: 5571: 5567: 5561: 5559: 5557: 5555: 5553: 5551: 5549: 5547: 5545: 5538: 5532: 5530: 5528: 5526: 5524: 5522: 5520: 5518: 5516: 5514: 5512: 5503: 5499: 5495: 5491: 5487: 5483: 5476: 5472: 5463: 5460: 5458: 5455: 5454: 5448: 5446: 5442: 5438: 5419: 5412: 5408: 5400: 5397: 5394: 5391: 5388: 5385: 5382: 5379: 5372: 5371: 5370: 5357: 5354: 5351: 5348: 5345: 5342: 5334: 5331:Representing 5329: 5326: 5325: 5321: 5319: 5315: 5290: 5286: 5266: 5262: 5257: 5249: 5246: 5243: 5240: 5237: 5234: 5229: 5225: 5221: 5218: 5211: 5210: 5209: 5196: 5193: 5190: 5187: 5184: 5179: 5175: 5166: 5163:Representing 5141: 5137: 5117: 5113: 5108: 5100: 5097: 5092: 5088: 5084: 5079: 5075: 5071: 5066: 5062: 5054: 5053: 5052: 5050: 5045: 5044: 5040: 5038: 5034: 5030: 5020: 5003: 5000: 4997: 4994: 4991: 4988: 4981: 4980: 4979: 4962: 4959: 4956: 4953: 4950: 4947: 4944: 4941: 4938: 4931: 4930: 4929: 4912: 4909: 4906: 4903: 4900: 4897: 4894: 4888: 4885: 4882: 4872: 4871: 4870: 4853: 4850: 4847: 4844: 4841: 4838: 4835: 4832: 4825: 4824: 4823: 4806: 4803: 4800: 4797: 4794: 4791: 4784: 4783: 4782: 4778: 4777: 4773: 4756: 4753: 4750: 4747: 4744: 4741: 4734: 4733: 4732: 4715: 4712: 4709: 4706: 4703: 4700: 4697: 4694: 4691: 4688: 4681: 4680: 4679: 4662: 4659: 4656: 4653: 4647: 4644: 4641: 4631: 4630: 4629: 4612: 4609: 4606: 4603: 4600: 4593: 4592: 4591: 4588: 4585: 4584: 4580: 4576: 4573: 4572: 4568: 4547: 4543: 4535: 4529: 4523: 4520: 4514: 4511: 4505: 4502: 4496: 4490: 4487: 4481: 4478: 4472: 4469: 4466: 4463: 4460: 4457: 4454: 4451: 4448: 4445: 4442: 4439: 4436: 4433: 4430: 4427: 4420: 4419: 4418: 4401: 4394: 4390: 4368: 4356: 4353: 4350: 4342: is even 4336: 4324: 4321: 4315: 4310: 4305: 4297: 4294: 4288: 4283: 4279: 4271: 4270: 4269: 4251: 4247: 4239: 4236: 4233: 4230: 4221: 4218: 4217: 4213: 4192: 4188: 4180: 4174: 4168: 4165: 4159: 4153: 4150: 4144: 4138: 4135: 4132: 4129: 4126: 4123: 4120: 4117: 4114: 4111: 4108: 4105: 4098: 4097: 4096: 4093: 4090:They are all 4073: 4069: 4061: 4058: 4053: 4049: 4045: 4040: 4036: 4027: 4007: 4003: 3995: 3992: 3989: 3980: 3977: 3976: 3972: 3970: 3966: 3951: 3948: 3946: 3942: 3937: 3913: = 10a+9 3907: 3895: 3892: 3889: 3886: 3878: = 10a+7 3872: 3860: 3857: 3854: 3851: 3848: 3840: = 10a+3 3834: 3822: 3819: 3816: 3813: 3810: 3802: = 10a+1 3796: 3784: 3781: 3778: 3775: 3772: 3766: 3761: 3755: 3749: 3740: 3738: 3734: 3730: 3726: 3721: 3719: 3715: 3711: 3707: 3703: 3699: 3695: 3691: 3687: 3683: 3679: 3675: 3671: 3661: 3658: 3654: 3650: 3646: 3642: 3637: 3627: 3624: 3619: 3617: 3613: 3603: 3595: 3592: 3578: 3573: 3565: 3562: 3557: 3554: 3550: 3529: 3524: 3516: 3513: 3510: 3507: 3498: 3494: 3490: 3486: 3482: 3478: 3474: 3470: 3461: 3457: 3453: 3449: 3440: 3436: 3428: 3415: 3410: 3400: 3396: 3390: 3382: 3379: 3371: 3366: 3363: 3360: 3356: 3352: 3347: 3343: 3337: 3333: 3327: 3322: 3319: 3316: 3312: 3308: 3297: 3293: 3287: 3283: 3279: 3276: 3273: 3268: 3265: 3262: 3258: 3252: 3248: 3236: 3222: 3217: 3209: 3206: 3203: 3200: 3191: 3190: 3186: 3172: 3169: 3149: 3129: 3120: 3117: 3104: 3101: 3098: 3095: 3092: 3089: 3086: 3083: 3080: 3077: 3074: 3071: 3068: 3065: 3062: 3059: 3056: 3053: 3045: 3042: 3041: 3032: 3014: 3004: 3001: 2998: 2995: 2991: 2984: 2976: 2966: 2963: 2960: 2957: 2953: 2947: 2944: 2940: 2931: 2926: 2923: 2920: 2916: 2907: 2892: 2879: 2876: 2873: 2870: 2866: 2862: 2857: 2854: 2851: 2848: 2844: 2838: 2835: 2831: 2822: 2817: 2814: 2811: 2807: 2798: 2783: 2770: 2767: 2764: 2761: 2757: 2753: 2745: 2742: 2739: 2736: 2732: 2726: 2723: 2719: 2710: 2705: 2702: 2699: 2695: 2683: 2670: 2665: 2653: 2649: 2643: 2639: 2635: 2632: 2629: 2624: 2621: 2618: 2615: 2611: 2605: 2602: 2598: 2586: 2564: 2560: 2554: 2550: 2546: 2543: 2540: 2535: 2532: 2529: 2526: 2522: 2516: 2513: 2509: 2496: 2493: 2490: 2487: 2484: 2481: 2478: 2475: 2472: 2469: 2468: 2464: 2461: 2458: 2457:For example: 2452: 2445: 2442: 2439: 2435: 2432: 2429: 2428: 2427: 2424: 2423: 2419: 2418: 2414: 2410: 2406: 2402: 2398: 2394: 2390: 2383: 2379: 2375: 2371: 2367: 2359: 2355: 2351: 2344: 2340: 2338: 2334: 2330: 2326: 2321: 2320: 2316: 2271: 2262: 2261: 2244: 2243: 2224: 2223: 2219: 2212: 2208: 2201: 2197: 2194: 2179: 2172: 2168: 2164: 2157: 2156: 2152: 2150: 2146: 2141: 2137: 2135: 2131: 2127: 2123: 2118: 2114: 2111: 2104: 2100: 2097: 2093: 2090: 2086: 2083: 2079: 2078: 2077: 2073: 2069: 2067: 2063: 2058: 2056: 2052: 2048: 2044: 2040: 2036: 2031: 2025: 2021: 2018: 2014: 2011: 2009: 2004: 2000: 1996: 1993: 1989: 1988: 1987: 1985: 1981: 1977: 1973: 1969: 1961: 1956: 1953: 1948: 1945: 1943: 1938: 1934: 1931: 1927: 1923: 1920: 1916: 1912: 1908: 1904: 1900: 1896: 1892: 1890: 1886: 1885: 1884: 1881: 1877: 1875: 1871: 1867: 1863: 1859: 1855: 1851: 1847: 1843: 1837: 1835: 1832: + 1831: 1827: 1824: + 1823: 1818: 1816: 1812: 1808: 1805: + 1804: 1799: 1796: 1792: 1788: 1785: + 1784: 1771: 1768: 1765: 1764: 1763: 1762: 1758: 1754: 1752: 1748: 1735: 1732: 1729: 1726: 1723: 1720: 1717: 1714: 1711: 1708: 1707: 1706: 1705: 1698: 1695: 1692: 1689: 1686: 1683: 1680: 1677: 1674: 1673: 1672: 1671: 1667: 1663: 1659: 1655: 1653: 1640: 1637: 1634: 1631: 1630: 1629: 1628: 1621: 1618: 1615: 1612: 1609: 1605: 1602: 1599: 1598: 1597: 1596: 1595:Second method 1589: 1586: 1583: 1580: 1577: 1574: 1573: 1572: 1571: 1567: 1563: 1559: 1556: 1544: 1541: 1538: 1535: 1532: 1529: 1528: 1527: 1526: 1522: 1520: 1516: 1512: 1507: 1505: 1500: 1488: 1485: 1482: 1479: 1476: 1473: 1472: 1471: 1470: 1466: 1447: 1444: 1443: 1439: 1436: 1435: 1431: 1428: 1427: 1423: 1420: 1418: 1417: 1412: 1408: 1405: 1404: 1400: 1397: 1396: 1392: 1389: 1388: 1384: 1381: 1377: 1373: 1369: 1365: 1361: 1357: 1353: 1351: 1350: 1345: 1341: 1338: 1336: 1335: 1330: 1326: 1323: 1322: 1318: 1315: 1314: 1310: 1307: 1306: 1302: 1299: 1297: 1296: 1291: 1287: 1284: 1283: 1279: 1276: 1274: 1273: 1268: 1264: 1261: 1259: 1258: 1253: 1249: 1246: 1244: 1243: 1238: 1234: 1231: 1230: 1226: 1223: 1222: 1218: 1215: 1214: 1210: 1207: 1205: 1204: 1199: 1195: 1192: 1190: 1189: 1184: 1180: 1177: 1176: 1172: 1169: 1168: 1164: 1161: 1157: 1153: 1149: 1145: 1141: 1137: 1133: 1131: 1130: 1125: 1121: 1118: 1117: 1113: 1110: 1109: 1105: 1102: 1100: 1099: 1094: 1090: 1087: 1086: 1082: 1079: 1075: 1071: 1067: 1063: 1059: 1055: 1051: 1047: 1043: 1041: 1040: 1035: 1031: 1028: 1026: 1025: 1020: 1016: 1013: 1009: 1005: 1001: 997: 993: 989: 985: 981: 977: 976: 972: 969: 968: 964: 961: 960: 956: 953: 951: 950: 945: 941: 938: 937: 933: 930: 929: 926: 922: 919: 918: 914: 911: 910: 906: 903: 901: 900: 895: 891: 888: 886: 885: 880: 874: 871: 870: 866: 863: 861: 860: 855: 851: 848: 847: 843: 840: 839: 835: 832: 831: 827: 824: 820: 818: 817: 812: 808: 805: 804: 800: 797: 795: 794: 789: 785: 782: 781: 777: 774: 773: 769: 766: 765: 761: 758: 757: 753: 750: 749: 745: 742: 740: 739: 734: 730: 727: 726: 722: 719: 717: 716: 711: 707: 704: 703: 699: 696: 694: 693: 688: 684: 681: 680: 676: 673: 672: 668: 665: 664: 660: 657: 656: 652: 649: 647: 646: 641: 638: 634: 630: 629: 625: 621: 620: 616: 613: 612: 609: 606: 602: 599: 595: 591: 587: 583: 579: 575: 571: 570: 566: 563: 562: 558: 555: 554: 550: 547: 546: 542: 539: 535: 533: 532: 527: 523: 508: 505: 502: 478: 474: 466: 463: 458: 454: 445: 444: 440: 437: 435: 434: 429: 425: 422: 420: 419: 414: 410: 407: 406: 402: 397: 396: 392: 389: 387: 386: 381: 377: 374: 373: 369: 366: 365: 359: 356: 354: 353: 348: 344: 341: 339: 338: 333: 329: 326: 324: 323: 318: 314: 311: 308: 307: 298: 295: 293: 290: 288: 285: 283: 280: 278: 275: 273: 270: 268: 265: 263: 260: 258: 255: 253: 250: 248: 245: 243: 240: 238: 235: 233: 230: 228: 225: 223: 220: 218: 215: 213: 210: 208: 205: 203: 200: 198: 195: 193: 190: 188: 185: 183: 180: 178: 175: 173: 170: 168: 165: 163: 160: 158: 155: 153: 150: 149: 146: 130: 125: 121: 115: 110: 106: 100: 95: 91: 81: 79: 75: 70: 67: 65: 54: 52: 51: 46: 42: 38: 34: 30: 26: 22: 5876:cut-the-knot 5847: 5825: 5800: 5781:Oeis A333448 5780: 5770: 5760: 5755: 5747: 5742: 5731:. Retrieved 5727:the original 5720: 5712: 5693: 5686: 5678: 5675:the original 5669: 5662: 5565: 5485: 5481: 5475: 5444: 5440: 5436: 5434: 5332: 5330: 5327: 5323: 5322: 5317: 5313: 5311: 5164: 5162: 5048: 5046: 5042: 5041: 5036: 5032: 5026: 5018: 4977: 4927: 4868: 4821: 4779: 4775: 4774: 4771: 4730: 4677: 4627: 4589: 4586: 4582: 4581: 4577: 4574: 4570: 4569: 4566: 4416: 4374: is odd 4222: 4219: 4215: 4214: 4211: 3981: 3978: 3974: 3973: 3962: 3949: 3938: 3741: 3736: 3732: 3728: 3724: 3722: 3717: 3713: 3709: 3705: 3701: 3697: 3693: 3689: 3685: 3681: 3677: 3673: 3669: 3667: 3656: 3652: 3648: 3644: 3640: 3633: 3620: 3609: 3601: 3593: 3499: 3492: 3488: 3484: 3480: 3476: 3472: 3468: 3462: 3455: 3451: 3447: 3441: 3437: 3434: 3237: 3192: 3188: 3187: 3121: 3118: 3046: 3043: 3039: 3038: 2908: 2799: 2684: 2587: 2497: 2494: 2491: 2488: 2485: 2482: 2479: 2476: 2474:1 mod 7 = 1 2473: 2470: 2466: 2465: 2462: 2459: 2456: 2425: 2421: 2420: 2416: 2415: 2408: 2404: 2400: 2396: 2392: 2388: 2381: 2380: 2373: 2369: 2365: 2357: 2353: 2349: 2342: 2341: 2336: 2332: 2328: 2324: 2322: 2318: 2317: 2272: 2263: 2259: 2258: 2241: 2240: 2221: 2220: 2216: 2209: 2205: 2198: 2195: 2192: 2176: 2169: 2162: 2161: 2154: 2153: 2148: 2144: 2139: 2138: 2133: 2129: 2125: 2121: 2116: 2115: 2109: 2108: 2102: 2095: 2088: 2081: 2074: 2070: 2065: 2061: 2059: 2054: 2050: 2046: 2042: 2038: 2034: 2032: 2029: 2023: 2016: 2007: 2006: 2002: 1998: 1991: 1983: 1975: 1971: 1967: 1965: 1959: 1951: 1941: 1940: 1936: 1929: 1925: 1918: 1914: 1910: 1906: 1902: 1898: 1894: 1888: 1882: 1878: 1873: 1869: 1865: 1861: 1857: 1853: 1849: 1845: 1841: 1838: 1833: 1829: 1825: 1821: 1819: 1814: 1810: 1806: 1802: 1800: 1794: 1790: 1786: 1782: 1780: 1760: 1759: 1755: 1744: 1724: 1721: 1715: 1712: 1703: 1702: 1690: 1687: 1681: 1678: 1669: 1665: 1664: 1660: 1656: 1651: 1649: 1627:Third method 1626: 1625: 1610: 1607: 1594: 1593: 1581: 1578: 1570:General rule 1569: 1565: 1564: 1560: 1557: 1553: 1524: 1523: 1508: 1501: 1497: 1480: 1477: 1468: 1467: 1463: 1414: 1379: 1375: 1371: 1367: 1363: 1359: 1355: 1347: 1332: 1293: 1270: 1255: 1240: 1201: 1186: 1159: 1155: 1151: 1147: 1143: 1139: 1135: 1127: 1096: 1077: 1073: 1069: 1065: 1061: 1057: 1053: 1049: 1045: 1037: 1022: 1011: 1007: 1003: 999: 995: 991: 987: 983: 979: 947: 924: 897: 882: 857: 814: 791: 736: 713: 690: 643: 636: 607: 604: 597: 593: 589: 585: 581: 577: 573: 529: 431: 416: 383: 350: 335: 320: 82: 77: 73: 71: 68: 63: 60: 48: 20: 18: 5043:For 2 or 5: 2492:And so on. 2337:digit pairs 1926:13264513... 1519:base-twelve 1517:; thus, in 536:Forming an 5891:Categories 5746:Page 274, 5733:2006-12-12 5537:p. 100–101 5468:References 3458:× −4 = −17 2382:Example 2: 2360:× 14 = 182 2343:Example 1: 1939:1 becomes 1362:) × 3 − 29 1142:) × 2 − 21 1052:) × 2 − 19 986:) × 2 − 17 653:624: 24. 5395:− 5389:≡ 5380:− 5346:⋅ 5247:≡ 5235:⋅ 5185:⋅ 5098:≡ 5085:⋅ 4998:⋅ 4992:− 4957:⋅ 4945:⋅ 4939:− 4907:⋅ 4895:⋅ 4886:− 4869:and then 4848:⋅ 4836:⋅ 4795:⋅ 4745:⋅ 4704:⋅ 4692:⋅ 4654:⋅ 4604:⋅ 4512:− 4479:− 4473:≡ 4467:⋅ 4455:⋅ 4443:⋅ 4431:⋅ 4351:− 4311:≡ 4295:− 4289:≡ 4237:− 4234:≡ 4139:≡ 4133:⋅ 4121:⋅ 4109:⋅ 4092:congruent 4059:≡ 4046:≡ 3993:≡ 3965:binomials 3762:≡ 3623:composite 3598:Beyond 30 3555:− 3542:and that 3514:− 3380:− 3357:∑ 3353:≡ 3313:∑ 3304:¯ 3266:− 3207:− 3204:≡ 3170:− 3130:− 3099:× 3081:− 3069:− 3057:− 3002:− 2985:× 2964:− 2917:∑ 2877:− 2863:× 2855:− 2808:∑ 2768:− 2754:× 2743:− 2696:∑ 2660:¯ 2622:− 2571:¯ 2533:− 2399:× 15) + ( 2376:× 0 = −42 2309:Sum = 33 1980:remainder 1504:remainder 821:Form the 315:Examples 5502:24936675 5451:See also 4364:if 4332:if 3903:if 3868:if 3830:if 3792:if 3672:, where 3467:Answer: 2207:Step A. 2149:Ekhādika 2145:Ekhādika 1761:Example. 1666:Example. 1566:Example. 1525:Example. 1114:480: 80 309:Divisor 5791:Sources 3941:A333448 3483:× 12 + 3475:× 10 + 3454:× −3 + 2407:× 18 + 2395:× 75 + 2372:× 20 + 2356:× 75 + 2045:, then 1469:Example 37:decimal 29:divisor 25:integer 5854: 5832: 5813: 5701: 5656:p. 101 5585:p. 108 5570:p. 105 5500: 5324:For 7: 5282: 5273: 5133: 5124: 3954:Proofs 3925: 3706:mq + t 3657:really 3636:modulo 3491:× 4 + 3487:× 3 + 3479:× 9 + 3471:× 1 + 3450:× 1 + 3040:Method 2422:Method 1749:) and 5498:JSTOR 4024:(see 3612:prime 3446:Ans: 2403:× 37 2391:× 15 2368:× 18 2352:× 15 2035:a+b=c 2001:. So 1905:into 1515:radix 1080:is.) 1014:is.) 33:radix 5852:ISBN 5830:ISBN 5811:ISBN 5699:ISBN 4428:1000 3945:OEIS 3737:mq+t 3729:mq+t 3692:= 10 2132:+ 1× 2128:+ 0× 2124:+ 5× 2049:and 2037:and 1913:and 506:> 495:for 5874:at 5490:doi 5486:207 5443:− 2 5435:so 5409:mod 5335:as 5258:mod 5167:as 5109:mod 4601:100 4544:mod 4440:100 4391:mod 4248:mod 4189:mod 4106:100 4070:mod 4004:mod 3947:). 3943:in 3570:mod 3521:mod 3407:mod 3214:mod 3142:or 3011:mod 2973:mod 2889:mod 2780:mod 2666:mod 2405:− 3 2393:− 3 2370:− 3 2354:− 3 2105:×10 2098:×10 2091:×10 2084:×10 1992:116 1984:186 1974:to 1919:301 1901:or 1889:371 1753:. 1606:61 1382:.) 1370:+ 3 1162:.) 1150:+ 2 1146:= − 1068:+ 2 1060:+ 2 1002:+ 2 994:+ 2 990:= 3 600:.) 584:= 3 580:− 7 521:.) 475:mod 47:in 5893:: 5805:. 5779:. 5647:^ 5610:^ 5591:^ 5576:^ 5543:^ 5510:^ 5496:. 5484:. 5343:10 5320:. 5226:10 5176:10 5063:10 4883:21 4833:20 4792:10 4689:98 4642:98 4548:11 4452:10 4395:11 4280:10 4252:11 4231:10 4118:10 4037:10 3990:10 3696:+ 3621:A 3551:10 3508:10 3416:1. 3344:10 3201:10 3102:11 3090:22 2992:10 2867:10 2758:10 2585:. 2331:, 2329:−3 2327:, 2103:49 2096:35 2089:56 2082:14 2017:11 2003:11 1986:: 1937:30 1909:, 1897:, 1860:, 1856:, 1852:, 1848:, 1844:, 1688:11 1679:11 1582:92 1579:20 1478:37 1416:30 1378:+ 1366:= 1358:+ 1349:29 1334:28 1295:27 1272:26 1257:25 1242:24 1203:23 1188:22 1158:+ 1138:+ 1129:21 1098:20 1076:+ 1056:= 1048:+ 1039:19 1024:18 1010:+ 982:+ 949:17 899:16 884:15 859:14 816:13 793:12 738:11 715:10 596:+ 588:+ 576:+ 455:10 297:30 292:29 287:28 282:27 277:26 272:25 267:24 262:23 257:22 252:21 247:20 242:19 237:18 232:17 227:16 222:15 217:14 212:13 207:12 202:11 197:10 53:. 19:A 5860:. 5838:. 5819:. 5783:. 5736:. 5723:" 5566:m 5504:. 5492:: 5445:z 5441:y 5437:x 5420:, 5416:) 5413:7 5406:( 5401:z 5398:2 5392:y 5386:x 5383:2 5358:, 5355:z 5352:+ 5349:y 5333:x 5318:z 5314:x 5296:) 5291:n 5287:5 5279:r 5276:o 5267:n 5263:2 5255:( 5250:z 5244:z 5241:+ 5238:y 5230:n 5222:= 5219:x 5197:, 5194:z 5191:+ 5188:y 5180:n 5165:x 5147:) 5142:n 5138:5 5130:r 5127:o 5118:n 5114:2 5106:( 5101:0 5093:n 5089:5 5080:n 5076:2 5072:= 5067:n 5049:n 5037:m 5033:m 5004:. 5001:b 4995:2 4989:a 4963:, 4960:b 4954:2 4951:+ 4948:a 4942:1 4913:. 4910:b 4904:2 4901:+ 4898:a 4892:) 4889:1 4880:( 4854:, 4851:b 4845:2 4842:+ 4839:a 4807:, 4804:b 4801:+ 4798:a 4757:, 4754:b 4751:+ 4748:a 4742:2 4716:, 4713:b 4710:+ 4707:a 4701:2 4698:+ 4695:a 4663:b 4660:+ 4657:a 4651:) 4648:2 4645:+ 4639:( 4613:b 4610:+ 4607:a 4551:) 4541:( 4536:d 4533:) 4530:1 4527:( 4524:+ 4521:c 4518:) 4515:1 4509:( 4506:+ 4503:b 4500:) 4497:1 4494:( 4491:+ 4488:a 4485:) 4482:1 4476:( 4470:d 4464:1 4461:+ 4458:c 4449:+ 4446:b 4437:+ 4434:a 4402:. 4398:) 4388:( 4369:n 4357:, 4354:1 4337:n 4325:, 4322:1 4316:{ 4306:n 4302:) 4298:1 4292:( 4284:n 4255:) 4245:( 4240:1 4196:) 4193:3 4186:( 4181:c 4178:) 4175:1 4172:( 4169:+ 4166:b 4163:) 4160:1 4157:( 4154:+ 4151:a 4148:) 4145:1 4142:( 4136:c 4130:1 4127:+ 4124:b 4115:+ 4112:a 4077:) 4074:3 4067:( 4062:1 4054:n 4050:1 4041:n 4011:) 4008:3 4001:( 3996:1 3908:n 3896:, 3893:1 3890:+ 3887:a 3873:n 3861:, 3858:5 3855:+ 3852:a 3849:7 3835:n 3823:, 3820:1 3817:+ 3814:a 3811:3 3797:n 3785:, 3782:1 3779:+ 3776:a 3773:9 3767:{ 3759:) 3756:n 3753:( 3750:D 3733:m 3725:m 3718:D 3714:e 3710:D 3702:D 3698:q 3694:t 3690:N 3686:m 3682:D 3678:D 3674:D 3670:D 3653:x 3649:y 3645:x 3641:y 3579:3 3574:1 3566:4 3563:= 3558:1 3530:3 3525:1 3517:3 3511:= 3493:1 3489:2 3485:3 3481:4 3477:5 3473:6 3469:7 3456:3 3452:2 3448:1 3411:1 3401:i 3397:a 3391:i 3387:) 3383:1 3377:( 3372:n 3367:0 3364:= 3361:i 3348:i 3338:i 3334:a 3328:n 3323:0 3320:= 3317:i 3309:= 3298:0 3294:a 3288:1 3284:a 3280:. 3277:. 3274:. 3269:1 3263:n 3259:a 3253:n 3249:a 3223:1 3218:1 3210:1 3173:1 3150:+ 3105:, 3096:2 3093:= 3087:= 3084:1 3078:7 3075:+ 3072:0 3066:7 3063:+ 3060:0 3054:9 3020:) 3015:7 3005:2 2999:k 2996:2 2988:( 2982:) 2977:7 2967:1 2961:k 2958:2 2954:a 2948:k 2945:2 2941:a 2937:( 2932:n 2927:1 2924:= 2921:k 2893:7 2885:) 2880:2 2874:k 2871:2 2858:1 2852:k 2849:2 2845:a 2839:k 2836:2 2832:a 2828:( 2823:n 2818:1 2815:= 2812:k 2784:7 2776:] 2771:2 2765:k 2762:2 2751:) 2746:1 2740:k 2737:2 2733:a 2727:k 2724:2 2720:a 2716:( 2711:n 2706:1 2703:= 2700:k 2692:[ 2671:7 2654:1 2650:a 2644:2 2640:a 2636:. 2633:. 2630:. 2625:1 2619:n 2616:2 2612:a 2606:n 2603:2 2599:a 2565:1 2561:a 2555:2 2551:a 2547:. 2544:. 2541:. 2536:1 2530:n 2527:2 2523:a 2517:n 2514:2 2510:a 2409:2 2401:1 2397:2 2389:1 2387:( 2374:2 2366:1 2358:2 2350:1 2333:2 2325:1 2134:6 2130:2 2126:3 2122:1 2066:n 2062:n 2055:n 2051:c 2047:a 2043:n 2039:b 2026:. 2024:4 2019:. 2012:. 2010:6 2008:4 1999:4 1994:. 1976:6 1972:1 1968:0 1962:. 1960:0 1954:. 1952:7 1946:. 1944:1 1942:2 1932:. 1930:2 1915:2 1911:1 1907:0 1903:9 1899:8 1895:7 1874:2 1870:3 1866:1 1862:5 1858:4 1854:6 1850:2 1846:3 1842:1 1834:y 1830:x 1826:y 1822:x 1815:y 1811:x 1807:y 1803:x 1795:y 1791:x 1787:y 1783:x 1725:5 1722:8 1716:5 1713:8 1691:0 1682:0 1652:5 1611:4 1608:7 1481:6 1380:b 1376:a 1372:b 1368:a 1364:a 1360:b 1356:a 1160:b 1156:a 1152:b 1148:a 1144:a 1140:b 1136:a 1078:b 1074:a 1070:b 1066:a 1062:b 1058:a 1054:a 1050:b 1046:a 1012:b 1008:a 1004:b 1000:a 996:b 992:a 988:a 984:b 980:a 692:9 645:8 598:b 594:a 590:b 586:a 582:a 578:b 574:a 531:7 509:1 503:n 482:) 479:6 472:( 467:4 464:= 459:n 433:6 418:5 385:4 352:3 337:2 322:1 192:9 187:8 182:7 177:6 172:5 167:4 162:3 157:2 152:1 131:q 126:3 122:p 116:m 111:2 107:p 101:n 96:1 92:p 78:n 74:n 64:n

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