Problem
NT-B2-M04-P014 Composite Check
#14
★★★★☆ Level 4 of 5
Show that \(8!\not\equiv-1\pmod9\), and explain why this does not contradict Wilson.
The product \(8!\) contains factors \(3\) and \(6\).
Since \(8!\) contains factors \(3\) and \(6\), the product is divisible by \(9\). Thus \(8!\equiv0\pmod9\), not \(-1\). There is no contradiction: Wilson's theorem states \((p-1)!\equiv-1\pmod p\) for prime \(p\), while \(9\) is composite.
Useful anti-mistake: Wilson cannot be applied to a composite modulus.