Proof by Contradiction Exploration

Proposition If a, b ∈ Z and a ≥ 2, then a - b or a - (b +1). Proof. Suppose for the sake of contradiction there exit (a)________________________, and for which it is not true that a - b or a - (b +1). By De Morgan’s law, we have (b)________________________________. The definition of (c)_________________ says there are c, d ∈ Z with b = ac and b+1 = ad. Subtracting one equation from the other gives ad − ac = 1, so a(d − c) = 1. Since a is positive, d−c is also positive (otherwise a(d−c) would be negative). Then d − c is a positive integer and a(d − c) = 1, so a = 1/(d − c) < 2. Thus we have a ≥ 2 and a < 2, a (d)__________________.

Use the method of proof by contradiction to prove the following statement. Suppose n ∈ Z. If n² is odd, then n is odd.