question 4contrapositive proof Use the method of contraposit

question 4-contrapositive proof

Use the method of contrapositive proof to prove the following statements. (In each case you should also think about how a direct proof would work. You will find in most cases that contrapositive is easier.) Suppose n epsilon Z. If n^2 is even, then n is even.

Solution

assume that a does divide b and show that then a will also divide bc.

So, if a does divide b, then b = a*q for some integer q. Then b*c = (a*q) * c. This means that (b*c)/a = q*c is an integer, so a divides bc.