If a is an even integer and b is an odd integer then prove a − b is an odd integer

If a is an even integer and b is an odd integer then prove a − b is an odd integer

Let a be our even integer
Let b be our odd integer

We can express a = 2x (Standard form for even numbers) for some integer x
We can express b = 2y + 1 (Standard form for odd numbers) for some integer y

a - b = 2x - (2y + 1)
a - b = 2x - 2y - 1

Factor our a 2 from the first two terms:
a - b = 2(x - y) - 1

Since x - y is an integer, 2(x- y) is always even. Subtracting 1 makes this an odd number.