Express $413_7$ to base 5

Analyze the question: We need to convert a base 7 number to base 5. The easiest way is to go through base 10 first.

Step 1: Convert 413₇ to base 10

413₇ = 4×7² + 1×7¹ + 3×7⁰ = 4×49 + 7 + 3 = 196 + 7 + 3 = 206

Step 2: Convert 206 to base 5 (repeated division)

• 206 ÷ 5 = 41 remainder 1
• 41 ÷ 5 = 8 remainder 1
• 8 ÷ 5 = 1 remainder 3
• 1 ÷ 5 = 0 remainder 1

Read the remainders from bottom to top: 1311₅

Check: 1×125 + 3×25 + 1×5 + 1 = 125 + 75 + 5 + 1 = 206 ✓

Answer: Option C, 1311₅.

Tip: To change from one non-decimal base to another, go through base 10: expand using place values, then divide repeatedly by the new base and read the remainders upward. Always check by expanding your answer back to base 10.