The unit's digit in 13^61 is

The correct answer is Option (2) → 3

The unit's digit of \(13^{61}\) is the same as that of \(3^{61}\). The unit's digits of the powers of 3 repeat in a cycle of 4:

    \(3^1 = 3,\quad 3^2 = 9,\quad 3^3 = 27,\quad 3^4 = 81\)

so the unit's digits run 3, 9, 7, 1, 3, 9, 7, 1, …

    \(61 = 4 \times 15 + 1\)

The remainder is 1, so the unit's digit is that of \(3^1\), which is 3.