4. Prove that g.c.d. {p(m), o (n)} > 1 unless either m or n equals 2 or 1. [Hint: Use Exercise 4 of Section 6-1.] Exercise 4 of Section 6.1 is here for reference: 4. Suppose that f(m) = 0 (mod p) and that ptf'(m), where p is a prime. Use Exercise 3 to prove that there exists an r (unique modulo p) such that f(m + rp') = 0 (mod p²+¹).

12. Determine which of the following functions are one-to-one. (a) r:Z36 → Z36; r(n) = 5n (mod 36). (b) s: Z10 → Z10; s(n) = n+5 (mod 10). (c) t: Z10 → Z10; t(n) = 3n + 5 (mod 10).

2.18 Show that if p is a positive integer such that both p and p² + 2 are prime, then p = 3.

Let n > 1 be an integer. Prove that ℤ /n has exactly n elements. Hint 1: Use the following strategy: 1) Prove that ℤ /n = {[0], [1], · · · , [n − 1]}. (This shows that |ℤ /n| ≤ n.) 2) Prove that [i] 6= [j] whenever 0 ≤ i < j ≤ n − 1. (This shows that [0], · · · , [n − 1] are all distinct, hence |ℤ /n| ≥ n.)