Prove that the difference of two odd integers is even Give a

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

Solution

• Suppose n and m are odd integers.

• Then n = 2k + 1 and m = 2l + 1 for some k, l Z, by the definition of an odd integer.

• Therefore n - m = (2k + 1) - (2l + 1) = 2(k -l).

• Since k and l are integers, so is k - l . • Hence n - m = 2p with p = k - l Z.

• By the definition of an even integer, this shows that n - m is even.