Question

Prove that the square of any positive integer is of the form 4q or 4q + 1 for some integer q.

Answer 1

Since any positive integer n is of the form 2p or, 2p + 1

When n = 2p,

then n² = 4p² = 4a

where a = p²

When n = 2p + 1,

then n² = (2p + 1)² = 4p² + 4p + 1

⇒ 4p(p + 1) + 1 ⇒ 4m + 1

where m = p(p + 1)

Therefore square of any positive integer is of the form 4q or 4q + 1 for some integer q