Divisibility rules for numbers

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 

Divisor	Divisibility condition	Examples
1	No special condition. Any integer is divisible by 1.	2 is divisible by 1.
2	The last digit is even (0, 2, 4, 6, or 8).	1294: 4 is even.
3	Sum the digits. The result must be divisible by 3.	405 → 4 + 0 + 5 = 9 and 636 → 6 + 3 + 6 = 15 which both are clearly divisible by 3. 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 → 1 + 5 = 6, which is clearly divisible by 3.
	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.	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.
4	The last two digits form a number that is divisible by 4.	40,832: 32 is divisible by 4.
	If the tens digit is even, the ones digit must be 0, 4, or 8. If the tens digit is odd, the ones digit must be 2 or 6.	40,832: 3 is odd, and the last digit is 2.
	Twice the tens digit, plus the ones digit is divisible by 4.	40832: 2 × 3 + 2 = 8, which is divisible by 4.
5	The last digit is 0 or 5.	495: the last digit is 5.
6	It is divisible by 2 and by 3.	1458: 1 + 4 + 5 + 8 = 18, so it is divisible by 3 and the last digit is even, hence the number is divisible by 6.

those useful for numbers with fewer digits