1. **Problem statement:** Convert the number $124_5$ (base 5) to base 2 and base 6. 2. **Step 1: Convert from base 5 to base 10 (decimal).** The number $124_5$ means: $$1 \times 5^2 + 2 \times 5^1 + 4 \times 5^0$$ Calculate each term: $$1 \times 25 = 25$$ $$2 \times 5 = 10$$ $$4 \times 1 = 4$$ Sum these: $$25 + 10 + 4 = 39$$ So, $124_5 = 39_{10}$. 3. **Step 2a: Convert decimal 39 to base 2.** Divide 39 by 2 repeatedly and record remainders: - $39 \div 2 = 19$ remainder $1$ - $19 \div 2 = 9$ remainder $1$ - $9 \div 2 = 4$ remainder $1$ - $4 \div 2 = 2$ remainder $0$ - $2 \div 2 = 1$ remainder $0$ - $1 \div 2 = 0$ remainder $1$ Write remainders from last to first: $$100111_2$$ 4. **Step 2b: Convert decimal 39 to base 6.** Divide 39 by 6 repeatedly and record remainders: - $39 \div 6 = 6$ remainder $3$ - $6 \div 6 = 1$ remainder $0$ - $1 \div 6 = 0$ remainder $1$ Write remainders from last to first: $$103_6$$ **Final answers:** - (a) $124_5 = 100111_2$ - (b) $124_5 = 103_6$