1. **Problem Statement:** Find the first three common multiples of given sets of numbers. 2. **Formula and Explanation:** The common multiples of numbers are multiples that are shared by all numbers in the set. To find common multiples, first find the Least Common Multiple (LCM), then list multiples of the LCM. 3. **Part a: Numbers 2, 3, and 5** - Find LCM of 2, 3, and 5. - Prime factors: 2 = 2, 3 = 3, 5 = 5. - LCM = $2 \times 3 \times 5 = 30$. - First three common multiples: $30, 60, 90$. 4. **Part b: Numbers 3, 4, and 5** - Prime factors: 3 = 3, 4 = $2^2$, 5 = 5. - LCM = $2^2 \times 3 \times 5 = 60$. - First three common multiples: $60, 120, 180$. 5. **Part c: Numbers 3, 6, and 9** - Prime factors: 3 = 3, 6 = $2 \times 3$, 9 = $3^2$. - LCM = $2 \times 3^2 = 18$. - First three common multiples: $18, 36, 54$. 6. **Array and Common Multiples of 3 and 4:** - The array with 3 rows and 4 columns has $3 \times 4 = 12$ counters. - 12 is the first common multiple of 3 and 4. 7. **a. Arrays for other common multiples of 3 and 4:** - Other common multiples are multiples of 12: 24, 36, 48, 60, etc. - Arrays can be made with rows and columns that multiply to these numbers. 8. **b. Patterns noticed:** - Common multiples of 3 and 4 are multiples of 12. - Arrays for these multiples can be arranged in rows and columns divisible by 3 and 4 respectively. 9. **c. Explanation for 48, 60, and 40:** - 48 and 60 are multiples of 12, so they are common multiples of 3 and 4. - 40 is not a multiple of 12, so it is not a common multiple. 10. **Amari and Romano's baskets problem:** - They want baskets with balls that can be grouped equally by 4 or 5. - This means the number of balls must be a common multiple of 4 and 5. - LCM of 4 and 5 is $20$. - Four solutions: $20, 40, 60, 80$. 11. **Observation:** - The number of balls in each basket must be a multiple of 20 to satisfy both groupings. **Final answers:** - a) $30, 60, 90$ - b) $60, 120, 180$ - c) $18, 36, 54$ - Amari and Romano's baskets: $20, 40, 60, 80$