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, 3m or 3m + 2 for some integer m? Justify your answer.

Answer 1

No. 

Justification: 

By Euclid’s Division Lemma, 

a = bq + r, 0 ≤ r < b

Here, a is any positive integer and b = 3, 

⇒ a = 3q + r 

So, a can be of the form 3q, 3q + 1 or 3q + 2. 

Now, for a = 3q 

(3q)² = 9q² = 3m [where m = 3q²] 

for a = 3q + 1 

(3q + 1)² 

= 9q² + 6q + 1 

= 3(3q² + 2q) + 1 

= 3m + 1 [where m = 3q² + 2q] 

for a = 3q + 2 

(3q + 2)² = 9q² + 12q + 4 

= 9q² + 12q + 3 + 1 

= 3(3q² + 4q + 1) + 1 

= 3m + 1 [where m = 3q² + 4q + 1] 

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