linguistic:paraphrasing
Divisibility rules for numbers 1–30

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.

To test the divisibility of a number by a power of 2 or a power of 5 (2 or 5, in which n is a positive integer), one only need to look at the last n digits of that number.

To test divisibility by any number expressed as the product of prime factors p 1 n p 2 m p 3 q {\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 = 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.