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

No, the square of any positive integer cannot be written in the form 3m + 2 where m is a natural number

Justification:

According to Euclid's division lemma,

A positive integer ‘a’ can be written in the form of bq + r

a = bq + r, where b, q and r are any integers,

For b = 3

a = 3(q) + r, where, r can be an integers,

For r = 0, 1, 2, 3……….

3q + 0, 3q + 1, 3q + 2, 3q + 3……. are positive integers,

(3q)² = 9q²

= 3(3q²)

= 3m  ......(Where 3q² = m)

(3q + 1)² = (3q + 1)²

= 9q² + 1 + 6q

= 3(3q² + 2q) + 1

= 3m + 1  .......(Where, m = 3q² + 2q)

(3q + 2)² = (3q + 2)²

= 9q² + 4 + 12q

= 3(3q² + 4q) + 4

= 3m + 4  .......(Where, m = 3q² + 2q)

(3q + 3)² = (3q + 3)²

= 9q² + 9 + 18q

= 3(3q² + 6q) + 9

= 3m + 9  ......(Where, m = 3q² + 2q)

Hence, there is no positive integer whose square can be written in the form 3m + 2 where m is a natural number.