246655 mod 9 = ?

Concept: Powers of an integer taken modulo a fixed number repeat in a periodic pattern. If ak ≡ 1 (mod m), the powers of a modulo m repeat with period (cycle length) k — so any exponent can be replaced by its remainder modulo k without changing the value modulo m.

Application: Finding the cycle length of 2 modulo 9:

1. 21 ≡ 2, 22 ≡ 4, 23 ≡ 8, 24 ≡ 7, 25 ≡ 5, 26 ≡ 1 (mod 9) — the cycle length is 6, since 26 is the first power to return to 1.

2. Reduce the exponent 46655 modulo the cycle length 6: 46655 = 6 × 7775 + 5, so 46655 ≡ 5 (mod 6).

3. Therefore 246655 ≡ 25 (mod 9) = 32 ≡ 5 (mod 9).

Cross-check: Using the decomposition 46655 = 3 × 15551 + 2, write 246655 = (23)15551 × 22 = 815551 × 4. Since 8 ≡ −1 (mod 9) and 15551 is odd, (−1)15551 = −1, giving −1 × 4 = −4 ≡ 5 (mod 9) — the same result as above, confirming the remainder.

Reference working: