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

Finding the remainders of powers of 5 when divided by 7:

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

$5^2 = 25 \equiv 4 \pmod{7}$

$5^3 = 125 \equiv 6 \pmod{7}$

$5^4 = 625 \equiv 2 \pmod{7}$

$5^5 = 3125 \equiv 3 \pmod{7}$

$5^6 = 15625 \equiv 1 \pmod{7}$

The remainders repeat every 6 powers since $5^6 \equiv 1 \pmod{7}$.

Since the remainders repeat every 6 powers:

$60 \div 6 = 10$ with remainder $0$

$5^{60} = (5^6)^{10}$

$5^{60} = (5^6)^{10}$

$\equiv (1)^{10} \pmod{7}$

$= 1$

The remainder when $5^{60}$ is divided by 7 is $1$.