Problem
NT-B1-M01-P017 Prime Square Minus One
#17
★★★☆☆ Level 3 of 5
Prove that if \(p>3\) is prime, then \(24\mid p^2-1\).
Factor \(p^2-1=(p-1)(p+1)\).
Since \(p>3\) is prime, it is odd and not divisible by \(3\). The numbers \(p-1\) and \(p+1\) are two consecutive even numbers; one is divisible by \(4\), the other by \(2\), so their product is divisible by \(8\). Also, among \(p-1,p,p+1\), one is divisible by \(3\), but \(p\) is not, so \(3\mid(p-1)(p+1)\). Since \(8\) and \(3\) are coprime, \(24\mid p^2-1\).
A key template for primes \(p>3\).