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Divisibility Rules 2-13

by Eugene Chen of the Amador Valley Math Team

Divisibility rules are very important to math competitions. Instead of performing lots of long division to find prime factors, we can use divisibility rules to narrow down our search. In most math competitions, you will usually need to know divisibility rules up to 11. This lesson includes divisibility rules up to 13.

\bullet An integer is divisible by 2 if and only if the last digit of it is 0, 2, 4, 6, or 8. For example, 432 is divisible by 2 because its last digit is 2, but 575 is not divisible by 2 because its last digit is 5.

\bullet An integer is divisible by 3 if and only if the sum of its digits is divisible by 3. For example, 138 is divisible by 3 because the sum of its digits is 1 + 3 + 8 = 12, which is divisible by 3, but 457 is not divisible by 3 because the sum of its digits is 4 + 5 + 7 = 16, which is not divisible by 3.

\bullet An integer is divisible by 4 if the number formed by the last two digits of it is divisible by 4. For example, 344 is divisible by 4, because the number formed by the last two digits of it (44) is divisible by 4, but 754 is not divisible by 4, because the number formed by the last two digits of it (54) is not divisible by 4.

\bullet An integer is divisible by 5 if its last digit is 0 or 5. For example, 105 is divisible by 5 because it ends in 5, but 972 is not divisible by 5 because it ends in 2.

\bullet An integer is divisible by 6 if it satisfies the divisibility rules for both 2 and 3.

\bullet To determine if an integer is divisible by 7, take the last digit, double it, and subtract it from the number formed by the remaining digit. Continue this process until you reach a number that you know is or is not divisible by 7. If it is divisible by 7, the original number is. If it isn't divisible by 7, then the original number is not. For example, 756 is divisible by 7 because doubling the last digit and subtracting from the remaining digits yields 75 - 6(2) = 63, which is divisible by 7, but 354 is not divisible by 7 because doubling the last digit and subtracting from the remaining digits yields 35 - 2(4) = 27, which is not divisible by 7.

\bullet An integer is divisible by 8 if the number formed by its last three digits is divisible by 8. For example, 8432 is divisible by 8 because 432 is, but 254593 is not divisible by 8 because 593 is not divisible by 8.

\bullet An integer is divisible by 9 if the sum of its digits is divisible by 9. For example, 423 is divisible by 9 because the sum of its digits is 4 + 2 + 3 = 9, which is divisible by 9, but 853 is not divisible by 9 because the sum of its digits is 8 + 5 + 3 = 16, which is not divisible by 9.

\bullet An integer is divisible by 10 if its units digit is 0. For example, 850 is divisible by 10 because it ends in a 0, but 572 is not divisible by 10 because it ends in 2.

\bullet To determine if an integer is divisible by 11, alternately add and subtract the digits from left to right. If the resulting integer is divisible by 11, the original number is. For example, 726 is divisible by 11, because alternately adding and subtracting its digits yields 7 - 2 + 6 = 11, which is divisible by 11, but 864 is not divisible by 11, because alternately adding and subtracting its digits yields 8 - 6 + 4 = 6, which is not divisible by 11.

\bullet An integer is divisible by 12 if it satisfies the divisibility rules for 3 and 4.

\bullet To determine if an integer is divisible by 13, remove the last digit, multiply it by 9, and subtract this from the remaining digits. Continue this until you reach a number that you know is divisible or not divisible by 13. If it is divisible by 13, then the original number is divisible by 13, but if it is not divisible by 13, the original number is not. For example, 312 is divisible by 13, because multiplying the last digit by 9 and subtracting it from the rest of the number yields 31 - 9(2) = 13, which is divisible by 13, but 867 is not divisible by 13, because multiplying the last digit by 9 and subtracting it from the rest of the number yields 86 - 9(7) = 23, which is not divisible by 13.

Exercise 1: Determine if the following numbers are divisible by 2,3,4,\dots,13:

a) 17583

b) 86792

c) 98623

d) 98614

Problem 1: Prove the divisibility rules for 4 and 8.

Problem 2: Prove the divisibility rules for 7, 11, and 13.

Problem 1 Hint: Break up the number into two parts.
Problem 2 Hint: Prove that the operation in the divisibility rules does not change the modular residue.
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