1. **Problem Statement:** Write the multisets (bags) of prime factors of the numbers 320, 60, and 500. 2. **Prime Factorization Formula:** To find the prime factors of a number, divide the number by the smallest prime (2) repeatedly until it no longer divides evenly, then move to the next prime (3, 5, 7, ...). 3. **Step-by-step prime factorization:** i. 320: - Divide by 2: 320 ÷ 2 = 160 - Divide by 2: 160 ÷ 2 = 80 - Divide by 2: 80 ÷ 2 = 40 - Divide by 2: 40 ÷ 2 = 20 - Divide by 2: 20 ÷ 2 = 10 - Divide by 2: 10 ÷ 2 = 5 - 5 is prime - Multiset: $\{2, 2, 2, 2, 2, 2, 5\}$ ii. 60: - Divide by 2: 60 ÷ 2 = 30 - Divide by 2: 30 ÷ 2 = 15 - Divide by 3: 15 ÷ 3 = 5 - 5 is prime - Multiset: $\{2, 2, 3, 5\}$ iii. 500: - Divide by 2: 500 ÷ 2 = 250 - Divide by 2: 250 ÷ 2 = 125 - Divide by 5: 125 ÷ 5 = 25 - Divide by 5: 25 ÷ 5 = 5 - Divide by 5: 5 ÷ 5 = 1 - Multiset: $\{2, 2, 5, 5, 5\}$ 4. **Multiplicities of each element:** i. 320: 2 appears 6 times, 5 appears 1 time ii. 60: 2 appears 2 times, 3 appears 1 time, 5 appears 1 time iii. 500: 2 appears 2 times, 5 appears 3 times 5. **Cardinalities of each multiset:** i. 320: $6 + 1 = 7$ ii. 60: $2 + 1 + 1 = 4$ iii. 500: $2 + 3 = 5$