Greatest Common Factor Word Problems (GCF) worksheet - Free Printable

Problem 1: Mrs. Lovejoy's Flower Arrangements

#### Given:
- Red carnations: 36
- White carnations: 60
- Pink carnations: 72
- Each arrangement must have the same number of each color.
- We need to find the greatest number of arrangements she can make using all the carnations.

#### Solution:
To determine the greatest number of arrangements, we need to find the greatest common divisor (GCD) of the numbers 36, 60, and 72. The GCD will tell us the maximum number of arrangements that can be made such that each arrangement has the same number of each color of carnation.

1. Find the prime factorization of each number:
- \( 36 = 2^2 \times 3^2 \)
- \( 60 = 2^2 \times 3 \times 5 \)
- \( 72 = 2^3 \times 3^2 \)

2. Identify the common prime factors and their lowest powers:
- The common prime factors are 2 and 3.
- The lowest power of 2 in the factorizations is \( 2^2 \).
- The lowest power of 3 in the factorizations is \( 3^1 \).

3. Calculate the GCD:
\[
\text{GCD} = 2^2 \times 3^1 = 4 \times 3 = 12
\]

So, the greatest number of arrangements Mrs. Lovejoy can make is 12.

#### Determine the number of flowers in each arrangement:
- Total red carnations: 36
- Total white carnations: 60
- Total pink carnations: 72

Since there are 12 arrangements, divide each total by 12:
- Red carnations per arrangement: \( \frac{36}{12} = 3 \)
- White carnations per arrangement: \( \frac{60}{12} = 5 \)
- Pink carnations per arrangement: \( \frac{72}{12} = 6 \)

#### Final Answer for Problem 1:
\[
\boxed{12, 3, 5, 6}
\]

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Problem 2: John Boy's Goodie Bags

#### Given:
- Pickled pig's feet: 24
- Pickled eggs: 12
- Each bag must contain the same number of each item.
- We need to find the greatest number of identical bags he can make using all the food.

#### Solution:
To determine the greatest number of identical bags, we need to find the greatest common divisor (GCD) of the numbers 24 and 12. The GCD will tell us the maximum number of bags that can be made such that each bag has the same number of pickled pig's feet and pickled eggs.

1. Find the prime factorization of each number:
- \( 24 = 2^3 \times 3 \)
- \( 12 = 2^2 \times 3 \)

2. Identify the common prime factors and their lowest powers:
- The common prime factors are 2 and 3.
- The lowest power of 2 in the factorizations is \( 2^2 \).
- The lowest power of 3 in the factorizations is \( 3^1 \).

3. Calculate the GCD:
\[
\text{GCD} = 2^2 \times 3^1 = 4 \times 3 = 12
\]

So, the greatest number of bags John Boy can make is 12.

#### Determine the number of items in each bag:
- Total pickled pig's feet: 24
- Total pickled eggs: 12

Since there are 12 bags, divide each total by 12:
- Pickled pig's feet per bag: \( \frac{24}{12} = 2 \)
- Pickled eggs per bag: \( \frac{12}{12} = 1 \)

#### Final Answer for Problem 2:
\[
\boxed{12, 2, 1}
\]

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Final Answers:

1. Number of Arrangements: 12, Number of flowers in each arrangement: Red: 3, White: 5, Pink: 6
2. Number of bags: 12, Number of items in each bag: Pig's Feet: 2, Eggs: 1