Evaluate \( 3241_{\text {five }}-1342_{\text {five }} \)

To evaluate \( 3241_{five} - 1342_{five} \), we can convert these numbers from base 5 to base 10, perform the subtraction, and then convert the result back to base 5. First, let's convert \( 3241_{five} \) to base 10: \[ 3241_{five} = 3 \times 5^3 + 2 \times 5^2 + 4 \times 5^1 + 1 \times 5^0 \] Calculating each term: \[ = 3 \times 125 + 2 \times 25 + 4 \times 5 + 1 \times 1 \] \[ = 375 + 50 + 20 + 1 = 446 \] Next, convert \( 1342_{five} \) to base 10: \[ 1342_{five} = 1 \times 5^3 + 3 \times 5^2 + 4 \times 5^1 + 2 \times 5^0 \] Calculating each term: \[ = 1 \times 125 + 3 \times 25 + 4 \times 5 + 2 \times 1 \] \[ = 125 + 75 + 20 + 2 = 222 \] Now, perform the subtraction in base 10: \[ 446 - 222 = 224 \] Finally, we will convert \( 224_{10} \) back to base 5. We do this by repeatedly dividing by 5 and noting the remainders: 1. \( 224 \div 5 = 44 \) remainder \( 4 \) 2. \( 44 \div 5 = 8 \) remainder \( 4 \) 3. \( 8 \div 5 = 1 \) remainder \( 3 \) 4. \( 1 \div 5 = 0 \) remainder \( 1 \) Reading the remainders from bottom to top gives us \( 1344_{five} \). Thus, the result of \( 3241_{five} - 1342_{five} \) is \[ \boxed{1344_{five}} \]