Problem
NT-B2-M04-P010 The Factorial \((p-2)!\)
#10
★★★☆☆ Level 3 of 5
Let \(p\) be an odd prime. Prove that \((p-2)!\equiv1\pmod p\).
Write \((p-1)!=(p-1)(p-2)!\).
By Wilson, \((p-1)!\equiv-1\pmod p\). But \((p-1)!\equiv(p-1)(p-2)!\equiv-(p-2)!\pmod p\). Hence \(-(p-2)!\equiv-1\), so \((p-2)!\equiv1\pmod p\).
A very common consequence of Wilson.