Question

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

• A. 2
• B. 3
• C. 6
• D. 5

Answer 1

The Correct Option is C

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.