1. **Problem Statement:** Simplify each expression using modular arithmetic. 2. **Formula and Explanation:** The expression $a \mod m$ means the remainder when $a$ is divided by $m$. 3. **Calculations:** (a) $12 \mod 5$: Divide 12 by 5, quotient is 2, remainder is $12 - 5 \times 2 = 2$. (b) $25 \mod 4$: Divide 25 by 4, quotient is 6, remainder is $25 - 4 \times 6 = 1$. (c) $12 \mod 3$: Divide 12 by 3, quotient is 4, remainder is $12 - 3 \times 4 = 0$. (d) $58 \mod 7$: Divide 58 by 7, quotient is 8, remainder is $58 - 7 \times 8 = 2$. (e) $5 \mod 6$: Since 5 is less than 6, remainder is 5. 4. **Final Answers:** (a) 2 (b) 1 (c) 0 (d) 2 (e) 5