A number leaves remainder 3 when divided by 6. What is the remainder when its square is divided by 6?

Given data

• n ≡ 3 (mod 6)

Concept / Approach

• Square the congruence to obtain n^2 modulo 6.

Step-by-step calculation

n^2 ≡ 3^2 = 9 ≡ 3 (mod 6)

Insight

If n is of the form 6k + 3 (i.e., odd multiple of 3), then n^2 ≡ 3 (mod 6) always.

Final Answer

Remainder = 3.