1. let p be a prime number. Show that (p - 1)! is congruent to (-1)mod p
2. prove criteria for divisibility by 2, 3, 4, 5, 9, 10, 11 using congruences modulo appropriate powers of 10
for this one, I have proved 2 and 10 modulo 10, but am stuck on 3
for divisibility by 3:
say n = d0 + 10d1 + ... +(10^k)dk
so I need to prove that if d0 + d1 + ... +dk is divisible by 3, then so is n
I can see how it would work modulo 3 (start with d0 +d1 +... +dk is congruent to 0 mod 3 and manipulate it to get n is congruent to 0 mod 3), but the problem clearly states i need to use modulo appropriate powers of 10... a starting point on this would be great. Thanks!