20. 432 sandwiches and 240 icecreams were shared equally among all the people present. What is the largest number of people that could have been present? \( \begin{array}{llll}\text { (A) } 5 & \text { (B) } 9 & \text { (C) } 40 & \text { (D) } 44 \\ \text { (E) } 48\end{array} \)

To find the largest number of people that could have been present, we need to determine the greatest common divisor (GCD) of the two quantities: 432 sandwiches and 240 ice creams. The GCD will give us the largest number of people that can equally share both items. ### Step 1: Find the prime factorization of each number. **For 432:** - Divide by 2: \( 432 \div 2 = 216 \) - Divide by 2: \( 216 \div 2 = 108 \) - Divide by 2: \( 108 \div 2 = 54 \) - Divide by 2: \( 54 \div 2 = 27 \) - Divide by 3: \( 27 \div 3 = 9 \) - Divide by 3: \( 9 \div 3 = 3 \) - Divide by 3: \( 3 \div 3 = 1 \) So, the prime factorization of 432 is: \[ 432 = 2^4 \times 3^3 \] **For 240:** - Divide by 2: \( 240 \div 2 = 120 \) - Divide by 2: \( 120 \div 2 = 60 \) - Divide by 2: \( 60 \div 2 = 30 \) - Divide by 2: \( 30 \div 2 = 15 \) - Divide by 3: \( 15 \div 3 = 5 \) - Divide by 5: \( 5 \div 5 = 1 \) So, the prime factorization of 240 is: \[ 240 = 2^4 \times 3^1 \times 5^1 \] ### Step 2: Find the GCD using the prime factorizations. The GCD is found by taking the lowest power of each prime factor present in both factorizations. - For \(2\): The minimum power is \(2^4\). - For \(3\): The minimum power is \(3^1\). - For \(5\): It is not present in 432, so we do not include it. Thus, the GCD is: \[ GCD = 2^4 \times 3^1 = 16 \times 3 = 48 \] ### Conclusion The largest number of people that could have been present is \(48\). Thus, the answer is \( \text{(E) } 48 \).