A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1, i.e., 3m or 3m + 2 for some integer m? Justify your answer.

Question

Philoid · Accepted Answer

No

By Euclid’s Lemma, b = a × q + r, 0 ≤ r < a

Here, b is any positive integer

According to the question, a = 3 ⇒ b = 3q + r for 0 r < 3

So, this must be in the form 3 q, 3q + 1 or 3 q + 2.

and (3 q + 1)² = 9q² + 6q + 1

⇒ (3 q + 1)² = 3 (3q² + 2q) + 1

⇒ (3 q + 1)² = 3m + 1

[where m = 3q² + 2q ]

Hence, square of a positive integer of the form 3q + 1 is always in the form 3m + 1 for some integer m.