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Meg is putting bottles and cans in bags to take to a recycling facility. She puts bottles in bags of 8 and cans in bags of 7, with no partially filled bags. While counting up the cans and bottles, an employee at the recycling facility points out that Meg has the same number of each. What is the smallest number of bottles that she can recycle?

Question

Meg is putting bottles and cans in bags to take to a recycling facility. She puts bottles in bags of 8 and cans in bags of 7, with no partially filled bags. While counting up the cans and bottles, an employee at the recycling facility points out that Meg has the same number of each. What is the smallest number of bottles that she can recycle?

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Solution

To solve this problem, we need to find the least common multiple (LCM) of 8 (the number of bottles per bag) and 7 (the number of cans per bag). The LCM of two numbers is the smallest number that is a multiple of both numbers.

Here are the steps to find the LCM of 8 and 7:

  1. List the multiples of each number:

    Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112...

    Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112...

  2. Find the smallest number that appears in both lists:

    The smallest number that is a multiple of both 8 and 7 is 56.

So, the smallest number of bottles that Meg can recycle, given that she has the same number of bottles and cans and that they are packed in bags of 8 and 7 respectively, is 56.

This problem has been solved

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