Prove that the difference of two odd integers is even. give a justification at each step.

Let a, b be two odd integers. Then both can be written as

a=2k-1, k is an integer

b=2m-1, m is an integer

So,

a-b=2k-1 - (2m-1) = 2k-2m-2=2 (k-m-1)

Since, k and m and 1 are all integers, then (k-m-1) is also an integer. Hence, 2 (k-m-1) is an even integer.

Thus, a-b is even.