WAEC Mathematics 2023 Paper

Concept:To convert a number from one base to another, first convert it to base $10$, then repeatedly divide by the target base and read the remainders upward.Explanation:Start with $413_{7}$.$413_{7} = (4 \times 7^2) + (1 \times 7^1) + (3 \times 7^0)$.Compute: $4 \times 49 = 196$, $1 \times 7 = 7$, and $3 \times 1 = 3$.Add: $196 + 7 + 3 = 206$.So $413_{7} = 206_{10}$.Now convert $206_{10}$ to base $5$ by repeated division by $5$.$206 \div 5 = 41$, remainder $1$.$41 \div 5 = 8$, remainder $1$.$8 \div 5 = 1$, remainder $3$.$1 \div 5 = 0$, remainder $1$.Read the remainders from last to first to get $1311_{5}$.Answer:$413_{7} = 1311_{5}$, so the correct option is C.