Question

show that every positive integer is either even or odd.​

Answer 1

Answer:

let 1/2 a positive integer

and 3/2

then 1/2+3/2=4/2=2

similarly 1/2and 4/2

1/2+4/2=5/2 is odd, hence proved

Answer 2

Answer:

Let us assume that there exist a smallest positive integer that is neither odd nor even, say n. Since n is 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 (k+1) this implies n is even.

In both ways we have a contradiction.

Thus, every positive integer is either even or odd