Prove the statement below Use models of formal proofs For al

Prove the statement below. Use models of formal proofs.

For all integers a,b,and c,if a | b and a | c,then a |(b-c2) .

Allow a, b, and c be integers such that a|b and a|c. Then b = ak for some integer k and c = ......

Solution

The statement can be proved by contraposition.

Now proceeding through the contrapositive statement is:.

For all integers a, b, and c , if a | b then a | bc .

Now when we are trying to proof the last statement , let us assume that a, b, and c are arbitrary integers and a | b.

Then there must exists an integer k such that b = ka .

Through the conclusion we can write ,

bc = kac = (kc)a . Since kc is an integer, we have a | bc.