Question

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.

Answer 1

According to Euclid's lemma, b = aq + r, 0 ≤ r < a

Comparing aq + r with 3m + 2, a = 3 and 0 ≤ r < 3

This means that r = 0, 1 and 2.

Hence 3q + r where 0 ≤ r <3.

For r = 0,

3q + 0 = 3q

(3q)² = 9q² which is a form of 3m where m = 3q²

For r = 1,

(3q + 1)² = 9q² + 6q + 1

= 3(3q² + 2q) + 1

= 3m + 1

Where, m = 3q² + 2q

For r = 2,

(3q + 2)² = 9q² + 12q + 4

= 9q² + 12q + 3 + 1

= 3(3q² + 4q + 1) + 1

= 3m + 1

Where, m = 3q² + 4q + 1