What is the remainder when 4⁵ is divided by 15?

Concept

To find the remainder of a power on division by a number, use modular arithmetic: reduce the base modulo the divisor, then look for a repeating cycle of the powers. If a² ≡ 1 (mod m), then every even power of a is ≡ 1 and the powers cycle with period 2, so a high exponent collapses to a tiny calculation.

Application

1. Reduce the base: the base 4 is already smaller than the divisor 15, so 4 ≡ 4 (mod 15).

2. 4² = 16, and 16 = 15 + 1, so 4² ≡ 1 (mod 15). The square of the base leaves remainder 1.

3. Rewrite the exponent using this: 4⁵ = 4⁴ × 4 = (4²)² × 4.

4. Substitute the remainder: (4²)² × 4 ≡ 1² × 4 = 1 × 4 = 4 (mod 15).

5. So the remainder is 4.

Cross-check

Compute directly: 4⁵ = 1024. Dividing, 15 × 68 = 1020, and 1024 − 1020 = 4. The direct division agrees, confirming the remainder is 4.