Prove there is no integer that is both even and odd

Let us take an integer x which is both even and odd.

• As an even integer, we write x in the form 2m for some integer m
• As an odd integer, we write x in the form 2n + 1 for some integer n

Since both the even and odd integers are the same number, we set them equal to each other
2m = 2n + 1

Subtract 2n from each side:
2m - 2n = 1

Factor out a 2 on the left side:
2(m - n) = 1

By definition of divisibility, this means that 2 divides 1.

But we know that the only two numbers which divide 1 are 1 and -1.

Therefore, our original assumption that x was both even and odd must be false.