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By coincidence, Trevor's Obedience School trained the same number of dogs and cats last week. The school teaches dogs in groups of 7 and cats in groups of 15. What is the smallest number of cats the school could have had?

Question

By coincidence, Trevor's Obedience School trained the same number of dogs and cats last week. The school teaches dogs in groups of 7 and cats in groups of 15. What is the smallest number of cats the school could have had?

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Solution

To find the smallest number of cats the school could have had, we need to find the least common multiple (LCM) of 7 and 15. The LCM is the smallest number that is a multiple of both numbers.

Here are the steps to find the LCM:

  1. List the prime factors of each number:

    • The prime factors of 7 are 7.
    • The prime factors of 15 are 3 and 5.

  2. Multiply each factor the greatest number of times it occurs in either number:

    • 7 occurs once in the factors of 7 and does not occur in the factors of 15. So, we multiply by 7 once.
    • 3 occurs once in the factors of 15 and does not occur in the factors of 7. So, we multiply by 3 once.
    • 5 occurs once in the factors of 15 and does not occur in the factors of 7. So, we multiply by 5 once.

  3. Multiply the results from step 2 together:

    • 7 * 3 * 5 = 105

So, the smallest number of cats the school could have had is 105.

This problem has been solved

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