The least non-negative remainder, when $5^{61}$ is divided by 7, is

5

The correct answer is Option (3) → 5 **

Compute the least non-negative remainder of $5^{61}$ modulo $7$.

Powers of $5$ modulo $7$:

$5^1 \equiv 5$

$5^2 \equiv 25 \equiv 4$

$5^3 \equiv 20 \equiv 6$

$5^4 \equiv 30 \equiv 2$

$5^5 \equiv 10 \equiv 3$

$5^6 \equiv 15 \equiv 1$

Cycle length = $6$.

$61 \mod 6 = 1$

$\Rightarrow 5^{61} \equiv 5^1 \equiv 5 \pmod{7}$

Least non-negative remainder = 5