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Stacy enjoys bird-watching and observed two types of birds travelling this season: ducks and seagulls. While the ducks travelled in flocks of 4, the seagulls travelled in flocks of 18. If Stacy observed the same total number of ducks and seagulls, what is the smallest number of ducks that she could have observed?

Question

Stacy enjoys bird-watching and observed two types of birds travelling this season: ducks and seagulls. While the ducks travelled in flocks of 4, the seagulls travelled in flocks of 18. If Stacy observed the same total number of ducks and seagulls, what is the smallest number of ducks that she could have observed?

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Solution

To solve this problem, we need to find the least common multiple (LCM) of the numbers 4 (the number of ducks in a flock) and 18 (the number of seagulls in a flock). 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 4 and 18:

  1. List the prime factors of each number:

    • The prime factors of 4 are 2 and 2.
    • The prime factors of 18 are 2, 3, and 3.
  2. For each prime number that appears in either list, count the number of times it appears in each list, and take the maximum count.

    • The prime number 2 appears twice in the list for 4 and once in the list for 18, so we take the maximum count, which is 2.
    • The prime number 3 does not appear in the list for 4, but it appears twice in the list for 18, so we take the maximum count, which is 2.
  3. Multiply together each prime number raised to the power of its maximum count. This gives us the LCM.

    • 2 raised to the power of 2 is 4.
    • 3 raised to the power of 2 is 9.
    • Multiplying 4 and 9 together gives us 36.

So, the LCM of 4 and 18 is 36. This means that the smallest number of ducks that Stacy could have observed is 36.

This problem has been solved

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