3 Name: Write or type solutions on a separate paper. If written, write legibly. 1. Provide a counterexample for the following statements. (a) If a and b are integers where a | b and b | a, then a = b. (b) If n is a real number where n2 > 0, then n > 0. (c) If n is an even integer, then n2 + 1 is prime. (d) If n is a positive integer, then n3 > n!. 2. Use any method to prove or disprove the following statements: (a) If n is a positive integer between 1 and 12 such that 6 | n, then 3 | n and 2 | n. (b) If m and n are positive integer such that gcd(m, n) = d, then d2 | m and d2 | n. (c) If m and n are positive integer such that gcd(m, n) = d, then d2 | mn. 3. Prove using induction principle: (a) n i2 = i=1 n n + 1 2n + 1 6 (b) If n 8 , then n = 5q + 3p where q and p are positive integers. (c) For n > 1 and x > 0 1+x n >1+xn