# Clue to Puzzle #15: Prime Squared Minus 1 Multiple of 24

15. Why is it that if 'p' is a prime number bigger than 3, then p2-1 is always divisible by 24 with no remainder?

Firstly we need to expand p2 - 1. If you don't know how to do that showing you how to do that may well be enough.

p2 - 1 = (p - 1) x (p + 1)

Secondly there is nothing special about the number 24 per se. The answer goes more like if I could show that something were a multiple of 2, 3 & 5 then that would show it were a multiple of 30.

2, 3, 5 are, in the example above, the prime factors of 30.

All prime numbers are, by definition, odd.

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