Prove by dirrect or contrapositive method if x and y are two

Prove by dirrect or contrapositive method....

if x and y are two integers whose product is odd, then both must be odd.

Solution

Proof by contradiction :

Given that xy is odd we assume that x and y are not both odd. If x and y are not both even then we must consider two cases.

Case 1: Let x be even and y is odd. By definition we have x = 2p and y = 2q + 1 where p, q are integers. Consider xy = (2p)(2q +1)= 4pq + 2p = 2(2pq +p) which is even. This is a contradicts that xy is in fact odd.

Case 2: Let x and y both be even. By definition x= 2p and y = 2q for p, q integers. Consider the product xy = (2p)(2q) = 2(2pq) = 2d for d = 2pq is an integer. There we have that xy is even which is a contradiction to the fact that xy is odd. There fore by the two cases above x and y must both be odd