The remainder, when 5^60 is divided by 7, is

The correct answer is Option (2) → 1

In modular arithmetic the quickest route is a power of 5 that leaves remainder \(\pm 1\) on division by 7:

    \( 5^3 = 125 = 7 \times 18 - 1 \)

    \( 5^3 \equiv -1 \pmod{7} \)

    \( 5^{60} = (5^3)^{20} \equiv (-1)^{20} = 1 \pmod{7} \)

So the remainder is 1. (By Fermat’s little theorem, \(5^6 \equiv 1 \pmod{7}\), and 60 is a multiple of 6, which gives the same result.)