What is the unit digit of (2153)167?

Concept: The unit digit of a power an depends only on the unit digit of a. The unit digits produced by successive powers of any digit repeat in a fixed cycle (of length 1, 2, or 4), so the unit digit of an equals the unit digit of the cycle's term at position r, where r is the remainder when n is divided by the cycle length (use r = cycle length when the division is exact).

Application:

1. The unit digit of the base 2153 is 3.

2. Powers of 3 repeat with cycle length 4: 31 = 3, 32 = 9, 33 = 7, 34 = 1, and the pattern then repeats.

3. Divide the exponent 167 by the cycle length 4: 167 = 4 × 41 + 3, so the remainder is 3.

4. A remainder of 3 means the unit digit matches the 3rd term of the cycle, 33 = 7.

Cross-check: Write 167 = 4 × 41 + 3. Considering only unit digits (i.e., working modulo 10): 3167 ≡ (34)41 × 33 ≡ 141 × 7 ≡ 7 (mod 10), since 34 = 81 has unit digit 1. This confirms the unit digit of (2153)167 is 7.