Question

The least non-negative remainder when 351 is divided by 7 is:

• A. 2
• B. 3
• C. 6
• D. 5

Answer 1

Answer: (C)

6

Answer 2

Use modular arithmetic to calculate $3^{51} \mod 7$. Note that:

$3^1 \equiv 3 \mod 7, \quad 3^2 \equiv 9 \equiv 2 \mod 7, \quad 3^3 \equiv 6 \mod 7, \quad 3^4 \equiv 18 \equiv 4 \mod 7.$

Observe that the powers of 3 modulo 7 repeat cyclically every 6 steps: 3, 2, 6, 4, 5, 1.

Since $51 \mod 6 = 3$, the equivalent power is $3^3$. From above:

$3^3 \equiv 6 \mod 7.$

Thus, the least non-negative remainder is 6.