What is the unit digit of (23)123456 ?

The unit digits of powers of 2 repeat in a cycle of length 4: 2, 4, 8, 6 (for exponents 1, 2, 3, 4), and this pattern repeats for every subsequent block of 4 exponents. So the unit digit of 2n depends only on the remainder when n is divided by 4 — remainder 1 → 2, remainder 2 → 4, remainder 3 → 8, and remainder 0 → 6.

1. Apply the power-of-a-power rule to combine the two exponents: (23)123456 = 2(3 × 123456).

2. Multiply the exponents: 3 × 123456 = 370368, so the expression becomes 2370368.

3. Divide the exponent by 4 to locate its position in the repeating cycle: 370368 ÷ 4 = 92592, remainder 0.

4. A remainder of 0 places the power at the 4th position in the cycle (2, 4, 8, 6), so the unit digit equals that of 24 = 16, which is 6.

Cross-check: 370368 is also exactly divisible by 8 (370368 ÷ 8 = 46296), so 2370368 shares its unit digit with 28 = 256 — again 6, confirming the result independently.

So, the unit digit of (23)123456 is 6.