Divisibility

A divisibility rule is a shorthand way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits. Although there are divisibility tests for numbers in any radix, or base, and they are all different, this article presents rules and examples only for decimal, or base 10, numbers. Martin Gardner explained and popularized these rules in his September 1962 "Mathematical Games" column in Scientific American.

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 the divisibility is obvious; for others (such as examining the last n digits) the result must be examined by other means.

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.

Note: To test divisibility by any number that can be expressed as 2ⁿ or 5ⁿ, in which n is a positive integer, just examine the last n digits.

Note: To test divisibility by any number expressed as the product of prime factors ${\displaystyle p_{1}^{n}p_{2}^{m}p_{3}^{q}}$, we can separately test for divisibility by each prime to its appropriate power. For example, testing divisibility by 24 (24 = 8*3 = 2³*3) is equivalent to testing divisibility by 8 (2³) and 3 simultaneously, thus we need only show divisibility by 8 and by 3 to prove divisibility by 24.

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