Question

Write whether the square of any positive integer can be of the form 3m + 2, where m is a natural number. Justify your answer.

Answer 1

Let a be any positive integer. Then by Euclid’s division lemma, we have 
a = bq + r, where 0 ≤ r < b 
For b = 3, we have 
a = 3q + r, where 0 ≤ r < 3 ...(i)

So, The numbers are of the form 3q, 3q + 1 and 3q + 2. 
So, (3q)2 = 9q2 = 3(3q2) 
= 3m, where m is a integer. 
(3q + 1)2 = 9q2 + 6q + 1 = 3(3q2 + 2q) + 1 
= 3m + 1, 
where m is a integer. 
(3q + 2)2 = 9q2 + 12q + 4, 
which cannot be expressed in the form 3m + 2. 
Therefore, Square of any positive integer cannot be expressed in the form 3m + 2.