11) Evaluate the following problem: 311five - 24five
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Transcribed Image Text:**Problem 11:**
Evaluate the following problem: \( 311_{five} - 24_{five} \).
This question involves subtraction in base 5 (quinary numeral system). To solve this, convert both numbers to base 10, perform the subtraction, and convert back to base 5 if needed.
1. **Convert to Base 10:**
- \( 311_{five} \):
- \( 3 \times 5^2 + 1 \times 5^1 + 1 \times 5^0 \)
- \( 3 \times 25 + 1 \times 5 + 1 \)
- \( 75 + 5 + 1 = 81 \)
- \( 24_{five} \):
- \( 2 \times 5^1 + 4 \times 5^0 \)
- \( 2 \times 5 + 4 \)
- \( 10 + 4 = 14 \)
2. **Subtract in Base 10:**
- \( 81 - 14 = 67 \)
3. **Convert Back to Base 5:**
- Divide 67 by 5 to convert back to base 5.
- \( 67 \div 5 = 13 \) remainder \( 2 \) (rightmost digit)
- \( 13 \div 5 = 2 \) remainder \( 3 \) (next digit)
- \( 2 \div 5 = 0 \) remainder \( 2 \) (leftmost digit)
- Result: \( 232_{five} \)
**Solution: \( 311_{five} - 24_{five} = 232_{five} \)**
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