Let p be an arbitrary prime number and let a be an arbitrary nonnegative integer. Prove the following fact 1. For any positive integer k, we have p | ak if and only if p | a. (Hint: use induction and...

Let p be an arbitrary prime number and let a be an arbitrary nonnegative integer. Prove the following fact

1. For any positive integer k, we have p | ak

if and only if p | a. (Hint: use induction and Lemma 7.12.)

2.For any integers n, m ∈ {1, . . . , p − 1}, we have that p 6 | nm.

3.For any integer m and any prime number q distinct from p (that is, p 6= q), we have m ≡p a andm ≡q a if and only if m ≡pq a. (Hint: think first about the case a = 0; then generalize.)

4.If 0 ≤ a <>∈ {1, p − 1}. (You may use Theorem 7.18 from p. 731