Found this in a Advanced GMAT Quant
Havent seen this anywhere else and would be interested in the proof for it if anyone can supply...
For all Prime numbers, p > 5
p^2 -1 is divisible by 24
A prime number greater than 3 is always of the form (6n+1) or (6n - 1) where n is a positive integer. Mind you, every number of the form (6n+1) or (6n - 1) is not prime but every prime (greater than 3) is of one of these forms. e.g. 5 = 6*1 - 1; 7 = 6*1+1; 11 = 6*2 - 1 etc
p^2 - 1 = (p-1)(p+1)
Since p is odd, (p is prime greater than 5), both (p-1) and (p+1) are even. Since (p-1) and (p+1) are consecutive even numbers, one of them will be divisible by 4. So (p-1)(p+1) has 8 as a factor.
Since p is of the form (6n+1) or (6n - 1), one of (p-1) and (p+1) must be of the form 6n i.e. it must be divisible by 6.
Hence, (p-1)(p+1) has 3 as a factor.
Therefore, (p-1)(p+1) has 8*3 = 24 as a factor.
These are not important rules/formulas. You can deduce these from other things you know but then there are unlimited properties you can deduce. So just focus on learning the basics.
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