Consider the following product. \[ 23_{\text {five }} \cdot 33_{\text {five }} \] Determine the product using base 5 . \( 23_{\text {five }} \cdot 33_{\text {five }}=\square \) five

To calculate the product \( 23_{\text{five}} \cdot 33_{\text{five}} \) in base 5, we first convert both numbers from base 5 to base 10. 1. **Convert \( 23_{\text{five}} \) to base 10:** \[ 23_{\text{five}} = 2 \cdot 5^1 + 3 \cdot 5^0 = 2 \cdot 5 + 3 \cdot 1 = 10 + 3 = 13_{\text{ten}} \] 2. **Convert \( 33_{\text{five}} \) to base 10:** \[ 33_{\text{five}} = 3 \cdot 5^1 + 3 \cdot 5^0 = 3 \cdot 5 + 3 \cdot 1 = 15 + 3 = 18_{\text{ten}} \] 3. **Multiply the two base 10 numbers:** \[ 13_{\text{ten}} \cdot 18_{\text{ten}} = 234_{\text{ten}} \] 4. **Convert \( 234_{\text{ten}} \) back to base 5:** We will divide \( 234 \) by \( 5 \) and keep track of the remainders. - \( 234 \div 5 = 46 \) remainder \( 4 \) - \( 46 \div 5 = 9 \) remainder \( 1 \) - \( 9 \div 5 = 1 \) remainder \( 4 \) - \( 1 \div 5 = 0 \) remainder \( 1 \) Now we read the remainders from bottom to top: \( 1, 4, 1, 4 \). Therefore, \( 234_{\text{ten}} = 1414_{\text{five}} \). Thus, the product \( 23_{\text{five}} \cdot 33_{\text{five}} \) in base 5 is: \[ \boxed{1414_{\text{five}}} \]