Question

Antonio is keeping an equal number of square and rectangular boxes in a local storage facility. he stores square boxes in stacks of 8 and rectangular boxes in stacks of 12. What is the smallest number of each type of box that Antonio could have?

Answer 1

The smallest number of square and rectangular boxes Antonio could have for each type, assuming he has an equal number of each, is 24, which is the least common multiple of 8 and 12.

Step-by-step explanation:

The student's question requires finding the smallest common multiple of the numbers 8 and 12, which represents the smallest number of boxes Antonio could have for each type, assuming he has an equal number of square and rectangular boxes. To find the smallest common multiple, we need to list the multiples of each number and identify the smallest multiple that appears in both lists, or we can find the least common multiple (LCM) by using the prime factors of each number.

The prime factors of 8 are 2 x 2 x 2, and the prime factors of 12 are 2 x 2 x 3. Combining these to find the LCM gives us 2 x 2 x 2 x 3 = 24. This means the smallest number of boxes that Antonio could have is 24 of each type.

Answer 2

The smallest number of square boxes Antonio could have is 24, and the smallest number of rectangular boxes is 12.

Step-by-step explanation:

To find the smallest number of each type of box, we need to determine the least common multiple (LCM) of 8 and 12. The LCM is the smallest number that both 8 and 12 divide into evenly.

The prime factorization of 8 is \(2^3\) and the prime factorization of 12 is
`\(2^2 * 3\).` The LCM is found by taking the highest power of each prime factor:
`\(2^3 * 3 = 24\).`

Therefore, the smallest number of square boxes (stacked in 8s) is 24, and the smallest number of rectangular boxes (stacked in 12s) is 12.

This ensures that Antonio can create stacks with an equal number of square and rectangular boxes, with the number of boxes in each stack being a multiple of both 8 and 12.