Question

For any integer m, square of the number is of the form ______ or ______

(a) 3m + 3, 3m – 2

(b) 3m – 2, 3m + 2

(c) 3m + 2, 3m – 3

(d) 3m, 3m + 1

I got this question in an online interview.

Enquiry is from Euclid’s Division Lemma in portion Real Numbers of Mathematics – Class 10

Answer 1

Correct option is (d) 3m, 3m + 1

Easiest explanation: Let a be any arbitrary number.

Then, by Euclid’s division lemma,

a = 3q + r where 0 ≤ r ≤ 3

a^2 = (3q + r)^2 = 9q^2 + r^2 + 6qr

When r = 0,

Then, a^2 = 9q^2 + 0^2 + 6q(0)

= 9q^2 = 3(3q^2) = 3m where m = 3q^2

Now, r = 1

a^2 = (3q+r)^2

= 9q^2 + (1)^2 + 6q(1)

= 9q^2 + 1 + 6q

= 3(3q^2 + 2q) + 1 = 3m + 1 where m = 3q^2 + 2q

When r = 2,

a^2 = (3q+r)^2

= 9q^2 + (2)^2 + 6q(2)

= 9q^2 + 12q + 4

= 3(3q^2) + 3(4)q + 3 × 1 + 1

= 3[(3q^2) + (4)q + 1] + 1

= 3m + 1 where m = (3q^2) + (4)q + 1

Hence, square of number n is of the form 3m or 3m + 1