Greatest Common Factor (GCF) Worksheets - Free Printable

Problem Analysis:

The image contains a series of word problems related to GCF (Greatest Common Factor) and LCM (Least Common Multiple). Each problem requires the application of these concepts to solve real-world scenarios. Below, I will solve each problem step by step.

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Problem 1:

A high-flying event manager conducts an event every 2 days. A local artist performs at his events every 3 days. If both are scheduled for today, how long will it be until they meet again at the same time?

#### Solution:
- The event manager conducts events every 2 days.
- The artist performs every 3 days.
- To find when they will meet again at the same time, we need to calculate the LCM (Least Common Multiple) of 2 and 3.

#### Steps:
1. Prime factorization:
- \(2 = 2\)
- \(3 = 3\)

2. LCM Calculation:
- The LCM is the product of the highest powers of all prime factors involved.
- Here, the prime factors are \(2\) and \(3\).
- LCM = \(2 \times 3 = 6\).

#### Answer:
They will meet again at the same time in 6 days.

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Problem 2:

Brian built a doghouse for his pet. He cut a piece of wood into 24 pieces of equal length and another piece of wood into 30 pieces of equal length. What is the greatest possible length of each piece?

#### Solution:
- Brian cuts one piece of wood into 24 pieces of equal length.
- He cuts another piece of wood into 30 pieces of equal length.
- To find the greatest possible length of each piece, we need to determine the GCF (Greatest Common Factor) of 24 and 30.

#### Steps:
1. Prime factorization:
- \(24 = 2^3 \times 3\)
- \(30 = 2 \times 3 \times 5\)

2. GCF Calculation:
- The GCF is the product of the lowest powers of all common prime factors.
- Common prime factors are \(2\) and \(3\).
- GCF = \(2^1 \times 3^1 = 6\).

#### Answer:
The greatest possible length of each piece is 6 units.

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Problem 3:

Scott wants to arrange identical flower pots on either side of a table so that there are no gaps between the flower pots. He has 18 white flower pots and 24 red flower pots. What is the greatest number of flower pots he can place on each side of the table?

#### Solution:
- Scott has 18 white flower pots and 24 red flower pots.
- To arrange them with no gaps, the number of flower pots on each side must be the same.
- We need to find the GCF (Greatest Common Factor) of 18 and 24 to determine the greatest number of flower pots per side.

#### Steps:
1. Prime factorization:
- \(18 = 2 \times 3^2\)
- \(24 = 2^3 \times 3\)

2. GCF Calculation:
- The GCF is the product of the lowest powers of all common prime factors.
- Common prime factors are \(2\) and \(3\).
- GCF = \(2^1 \times 3^1 = 6\).

#### Answer:
The greatest number of flower pots he can place on each side of the table is 6.

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Problem 4:

Its 9:00 p.m. right now. John’s alarm clock rings every 4 hours, and Jill’s alarm clock rings every 6 hours. At what time will their alarms ring together for the first time after 9:00 p.m.?

#### Solution:
- John’s alarm clock rings every 4 hours.
- Jill’s alarm clock rings every 6 hours.
- To find when their alarms will ring together, we need to calculate the LCM (Least Common Multiple) of 4 and 6.

#### Steps:
1. Prime factorization:
- \(4 = 2^2\)
- \(6 = 2 \times 3\)

2. LCM Calculation:
- The LCM is the product of the highest powers of all prime factors involved.
- Prime factors are \(2\) and \(3\).
- LCM = \(2^2 \times 3 = 4 \times 3 = 12\).

3. Time Calculation:
- Their alarms will ring together every 12 hours.
- Starting from 9:00 p.m., adding 12 hours gives 9:00 a.m. the next day.

#### Answer:
Their alarms will ring together at 9:00 a.m. the next day.

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Problem 5:

The factory sold an automobile every 3 minutes and a motorcycle every 5 minutes. If the factory sells automobiles and motorcycles at the same rate, how many minutes will it take for a car and a motorcycle to be sold at the same time?

#### Solution:
- An automobile is sold every 3 minutes.
- A motorcycle is sold every 5 minutes.
- To find when both an automobile and a motorcycle will be sold at the same time, we need to calculate the LCM (Least Common Multiple) of 3 and 5.

#### Steps:
1. Prime factorization:
- \(3 = 3\)
- \(5 = 5\)

2. LCM Calculation:
- The LCM is the product of the highest powers of all prime factors involved.
- Prime factors are \(3\) and \(5\).
- LCM = \(3 \times 5 = 15\).

#### Answer:
It will take 15 minutes for a car and a motorcycle to be sold at the same time.

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Final Answers:

1. \(\boxed{6}\) days
2. \(\boxed{6}\) units
3. \(\boxed{6}\) flower pots
4. \(\boxed{9:00 \text{ a.m.}}\)
5. \(\boxed{15}\) minutes

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These solutions are derived using the principles of GCF and LCM, ensuring clarity and correctness. Let me know if you have further questions!