1. **Problem Statement:** Simplify each number modulo the given modulus. 2. **Formula and Rules:** For any integer $a$ and modulus $m$, the modulo operation is defined as $a \bmod m = r$ where $r$ is the remainder when $a$ is divided by $m$, and $0 \leq r < m$. 3. **Step-by-step simplifications:** (i) $71 \bmod 6$: Divide 71 by 6: $71 = 6 \times 11 + 5$, remainder is 5. (ii) $125 \bmod 8$: Divide 125 by 8: $125 = 8 \times 15 + 5$, remainder is 5. (iii) $100 \bmod 12$: Divide 100 by 12: $100 = 12 \times 8 + 4$, remainder is 4. (iv) $134 \bmod 10$: Divide 134 by 10: $134 = 10 \times 13 + 4$, remainder is 4. (v) $67 \bmod 7$: Divide 67 by 7: $67 = 7 \times 9 + 4$, remainder is 4. (vi) $705 \bmod 11$: Divide 705 by 11: $705 = 11 \times 64 + 1$, remainder is 1. (vii) $12 \bmod 12$: Divide 12 by 12: $12 = 12 \times 1 + 0$, remainder is 0. (viii) $20 \bmod 9$: Divide 20 by 9: $20 = 9 \times 2 + 2$, remainder is 2. 4. **Final answers:** (i) 5 (ii) 5 (iii) 4 (iv) 4 (v) 4 (vi) 1 (vii) 0 (viii) 2 Each remainder is the simplified modulo result for the respective number and modulus.