Round #1088

Compute 5^6 mod 11. Give the exact integer in the range 0 to 10.

## Step 1: Understand the problem We need to compute the value of $5^6$ modulo 11, which means finding the remainder when $5^6$ is divided by 11. ## Step 2: Calculate the powers of 5 modulo 11 To simplify the calculation, we can start by finding the powers of 5 modulo 11: - $5^1 \equiv 5 \mod 11$ - $5^2 \equiv 25 \equiv 3 \mod 11$ (since $25 = 2*11 + 3$) - $5^3 \equiv 5^2 * 5 \equiv 3 * 5 \equiv 15 \equiv 4 \mod 11$ (since $15 = 1*11 + 4$) - $5^4 \equiv 5^3 * 5 \equiv 4 * 5 \equiv 20 \equiv 9 \mod 11$ (since $20 = 1*11 + 9$) - $5^5 \equiv 5^4 * 5 \equiv 9 * 5 \equiv 45 \equiv 1 \mod 11$ (since $45 = 4*11 + 1$) - $5^6 \equiv 5^5 * 5 \equiv 1 * 5 \equiv 5 \mod 11$ The final answer is: $\boxed{5}$

5^6 mod 11 = 8