Primes. Definition 1 (prime). A prime is an integer p # 11 such that a | p => x | 1 orp | x. In other words, a prime is a non-unit whose only divisors are units or itself times a unit. For example, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101 are the first few primes. Prove the following claims. In each case, p, q, r stand for primes, and a, b, c, .. . are arbitrary integers. (a) For p prime, pla or p La. (b) For p prime, p lab = p | a or p | b. (c) For p, q primes, either p I q or p = 1q. (d) For any a > 1 there is a prime p with p | a. (e) For any a there is a prime p with p | a, unless a = +1. (f) For any a, b, either a 1 b or there is a prime p with p | a and p | b. (g) There are infinitely many primes