3) Prove the following statement: “If x is an even integer, then x^2 is also even."

Question

Not sure if I solved it correctly, but feel free to solve it another way or correct me.

Here is what I did:

x^2=(2n+2)^2 -- I let nE(belongs to)R(all real number) such that x^2 is an even integer

=4n^2+4n+4

Thus, if x is an even integer, then x^2 is also even.

Answer 1

The underlying idea you used is correct. We may rewrite your proof as follows:

As x is an even integer, then x=2n for some integer n. Hence, x^2=(2n)^2=4n^2=2(2n^2). Since the multiplication of integers is an integer, then 2n^2= 2*n*n is an integer. As a result, x^2 is a multiple of 2 and hence it is also even.