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.