(a + b) % n = ((a % n) + (b % n)) % n
(a * b) % n = ((a % n) * (b % n)) % n
(a / b) % n = ((a % n) * ((b ^(-1)) % n)) % n
(a % n) % n = a % n
(identity)
(a * n) % n = 0
(as proven by multiplication above)
therefore (n!) % n = 0
This article has some nice modulus tricks.