Wilson's Theorem

Wilson's Theorem states that for a prime number $p$, the factorial $(p-1)!$ is congruent to $-1$ (or $p-1$) modulo $p$.

$(p-1)! \equiv -1 \pmod{p}$

This theorem is useful for problems asking for the remainder of a factorial when divided by a prime number. For example, for $6! \pmod{7}$, since 7 is prime, Wilson's Theorem applies directly: $6! \equiv -1 \equiv 6 \pmod{7}$.

If the modulus is not prime, you must check for common factors and simplify. For example, $4! = 24$, so $4! \equiv 0 \pmod{6}$.