Question

Show that every positive integer is either even or odd.

Answer 1

Let us assume that there exist a smallest positive integer that is neither odd nor even, say n. Since n is the least positive integer which is neither even nor odd, n − 1 must be either odd or even.

Case 1: If n − 1 is even, n − 1 = 2k for some k.
But this implies n = 2k + 1
this implies n is odd.

Case 2: If n − 1 is odd, n − 1 = 2k + 1 for some k.
But this implies n = 2k + 2 = 2(k + 1)
this implies n is even.

In both ways we have a contradiction.
Thus, every positive integer is either even or odd.