The remainder, when $5^{60}$ is divided by 7, is

1

The correct answer is Option (2) → 1 **

Compute $5^{60} \bmod 7$.

Since $5 \equiv -2 \pmod{7}$,

$5^{60} \equiv (-2)^{60} \pmod{7}$

$(-2)^{60} = 2^{60}$ (positive).

Now use the cycle of $2^{n} \pmod{7}$:

$2^{1}\equiv 2$

$2^{2}\equiv 4$

$2^{3}\equiv 8\equiv 1$

Cycle length is $3$.

$60 \bmod 3 = 0$

Therefore:

$2^{60} \equiv 2^{0} \equiv 1 \pmod{7}$

Remainder = $1$