5^100 (mod\ 9) =

The correct answer is Option (1) → 4

A small power of 5 that is −1 modulo 9 makes the large power easy:

    \(5^2 = 25 \equiv 7 \pmod 9\)
    \(5^3 = 125 = 13 \times 9 + 8 \equiv 8 \equiv -1 \pmod 9\)
    \(5^6 = \left(5^3\right)^2 \equiv (-1)^2 = 1 \pmod 9\)

Since 100 = 6 × 16 + 4:

    \(5^{100} = \left(5^6\right)^{16} \times 5^4 \equiv 1 \times 5^4 \pmod 9\)
    \(5^4 = 5^3 \times 5 \equiv (-1) \times 5 = -5 \pmod 9\)
    \(-5 \equiv -5 + 9 = 4 \pmod 9\)

So \(5^{100} \equiv 4 \pmod 9\).