2, 5 - If the last digit is 0, 2, 4, 6, or 8, the number is divisible by 2. If the last digit is 0 or 5, the number is divisible by 5.
3 - Add the digits together. Repeat until you have at most 2 digits. If the result is divisible by 3, so is the original number.
7, 13 - Alternately add and subtract the digits in groups of 3, then repeat until you reach a number with at most 3 digits. For example, 314153 is a multiple of 7 because 314-153=161=7*23 is divisible by 7, and 314158 is a multiple of 13 because 314-158=156 is divisible by 13.
11 - Alternately add and subtract the digits, like this: 4-1+9-8+8-6+2-2+7-2=11 is divisible by 11, so 4198862272 must also be a multiple of 11.
17, 5882353 - Alternately add and subtract the digits in groups of 8. If the result is divisible by 17 and/or 5882353, so is the original number.
19, 52579 - Alternately add and subtract the digits in groups of 9. If the result is divisible by 19 and/or 52579, so is the original number.
23, 4093, 8779 - Alternately add and subtract the digits in groups of 11. If the result is divisible by 23, 4093, and/or 8779, so is the original number. Also works for 121=11*11
29, 281, 121499449 - Alternately add and subtract the digits in groups of 14. If the result is divisible by 29, 281, and/or 121499449, so is the original number.
31, 2906161 - Add the digits in groups of 15. If the result is divisible by 31 and/or 2906161, so is the original number.