Course: Foundations of Higher Mathematics

Prove that if and only if

I feel confident about this one. It's only the first part of the biconditional. Here's my attempt.

Assume that , i.e. , for some integer a.

Then,

.

Since is an integer, b is an integer also.

Therefore is odd, which means that is odd, i.e. , for some integer c.

Then,

, for some integer d

Therefore , hence