I've blogged in the past about the little road-trip game of factoring license-plate numbers in one's head (see my archives). The primary tools are divisibility tests for the primes from 1 to 31. There are easy, schoolgirl tests for 2, 3, 5, and 11 base 10, 7 is not hard to just do, but 13, 17, 19, 21, 23, 29, and 31 are tougher. This last weekend, my wife came up with the following little NON-divisibility twist. Because these tougher primes divide NONE of the three-digit multiples of 100, we have the fact that if any of them divide the last two digits of a 3-digit number, then the number itself is NOT divisible by that prime. This is a consequence of the fact that if x divides (a+b), then x must divide a and b separately. So, for instance, since 51 = 3*17, we immediately know that 17 divides NONE of 151, 251, 351, ..., 951. Neat, huh? Definitely useful. She was kicking my butt at the game till I persuaded her to share her little secret.