GCF and LCM word problems worksheet for practicing greatest common factor and least common multiple in practical contexts.

Let's solve each of these GCF (Greatest Common Factor) and LCM (Least Common Multiple) word problems step by step.

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1) Berkley and Bronx take their Spanish lessons from a trainer in Jacksonville. Berkley goes to the class every 3 days, and Bronx attends the class once in 5 days. If they both took their lessons today, how long will it take Berkley and Bronx to take their lessons again on the same day?

Solution:
We are looking for the least common multiple (LCM) of 3 and 5 because we want to know when both will be attending class on the same day again.

- Multiples of 3: 3, 6, 9, 12, 15, ...
- Multiples of 5: 5, 10, 15, 20, ...

The LCM of 3 and 5 is 15.

✔ Answer: It will take 15 days for them to attend class on the same day again.

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2) Ethan has two rolls of jute rope. He measured them to be 78 feet and 91 feet long. If he wants to cut them into several pieces of equal lengths, what would be the greatest possible length of each piece?

Solution:
We need the greatest common factor (GCF) of 78 and 91 because we want the largest possible equal-length pieces that can divide both ropes evenly.

Use the Euclidean Algorithm:

- 91 ÷ 78 = 1 remainder 13
- 78 ÷ 13 = 6 remainder 0

So, GCF = 13

✔ Answer: The greatest possible length of each piece is 13 feet.

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3) Caryn and Delma are taking quizzes on various subjects. Caryn sets consecutive alarms on her watch to ring every 20 minutes, and Delma sets hers to ring every 15 minutes. If both girls start the quiz at the same time, how long will it take for their alarms to ring simultaneously?

Solution:
We need the LCM of 20 and 15.

Prime factorization:
- 20 = 2² × 5
- 15 = 3 × 5

LCM = 2² × 3 × 5 = 60

✔ Answer: Their alarms will ring simultaneously after 60 minutes.

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4) Beachcombing on a Sunday, Dylan found 30 conchs, 15 cowries, and 5 clamshells. If he’s planning to give them away in sets to his friends, how many identical sets can he make?

Solution:
To make identical sets, each set must have the same number of each type of shell. We need the GCF of 30, 15, and 5.

- Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
- Factors of 15: 1, 3, 5, 15
- Factors of 5: 1, 5

Common factors: 1 and 5 → GCF = 5

So, he can make 5 identical sets, each with:
- 30 ÷ 5 = 6 conchs
- 15 ÷ 5 = 3 cowries
- 5 ÷ 5 = 1 clamshell

✔ Answer: He can make 5 identical sets.

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5) Prepping up for a cycling event, Claudia, Evelyn, and Amanda covered the distance in 12 minutes, 15 minutes, and 18 minutes respectively. They had agreed that they’d continue to cycle in the same track until all three reach the finish line together. How many hours would the session continue for?

Solution:
We need the LCM of 12, 15, and 18 to find when all three will finish at the same time.

Prime factorizations:
- 12 = 2² × 3
- 15 = 3 × 5
- 18 = 2 × 3²

LCM = 2² × 3² × 5 = 4 × 9 × 5 = 180 minutes

Convert to hours:
180 ÷ 60 = 3 hours

✔ Answer: The session would continue for 3 hours.

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✔ Final Answers Summary:

1. 15 days
2. 13 feet
3. 60 minutes
4. 5 sets
5. 3 hours

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