The remainder, when 5^61 is divided by 8, is

The correct answer is Option (2) → 5

Since \( 5^2=25=3\times 8+1 \), the square of 5 leaves remainder 1 on division by 8:

    \( \displaystyle 5^2\equiv 1\ (\mathrm{mod}\ 8) \)

Then

    \( \displaystyle 5^{61}=5\times(5^2)^{30} \)

    \( \displaystyle \equiv 5\times 1^{30}=5\ (\mathrm{mod}\ 8) \)

So the remainder is 5.