Gadzooks

This is a given.

If the last digit is divisible by 2 (a.k.a. even), then so is the whole number.

If the sum of the digits is divisible by 3, then so is the whole number. The recursivity of this means that if the sum has multiple digits, you can add them up again until you get a single digit and see if it's 3, 6, or 9.

Like the rule for 2, but check if the last two digits are divisible by 4.

If it ends in 5 or 0.

If the rules for both 2 and 3 apply.

No shortcut. Alas.

Like the rules for 2 and 4, but check if the last three digits are divisible by 8. (Yes, this pattern keeps going for 16, 32, etc.)

Like the rule for 3, but the sum of the digits (or the sum of the sum of the digits, etc.) must be 9.