Grade 5 math worksheet focusing on GCF and LCM word problems with practical examples.

Problem 1: Gym Activities

Question: At the gym, Hillary swims every 6 days, runs every 4 days, and cycles every 16 days. If she did all three activities today, in how many days will she do all three activities again on the same day?

Solution:
To determine when Hillary will do all three activities (swimming, running, and cycling) on the same day again, we need to find the Least Common Multiple (LCM) of the intervals at which she performs each activity. The intervals are:
- Swimming: every 6 days
- Running: every 4 days
- Cycling: every 16 days

#### Step 1: Prime Factorization
- \( 6 = 2 \times 3 \)
- \( 4 = 2^2 \)
- \( 16 = 2^4 \)

#### Step 2: Identify the Highest Powers of All Prime Factors
The LCM is found by taking the highest power of each prime factor that appears in the factorizations:
- For \( 2 \): The highest power is \( 2^4 \) (from 16).
- For \( 3 \): The highest power is \( 3^1 \) (from 6).

#### Step 3: Calculate the LCM
\[
\text{LCM} = 2^4 \times 3^1 = 16 \times 3 = 48
\]

#### Final Answer:
Hillary will do all three activities on the same day again in 48 days.

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Problem 2: Packing CDs

Question: Oscar needs to ship 14 rock CDs, 12 classical CDs, and 8 pop CDs. He can pack only one type of CD in each box, and he must pack the same number of CDs in each box. What is the greatest number of CDs Oscar can pack in each box?

Solution:
To find the greatest number of CDs Oscar can pack in each box, we need to determine the Greatest Common Divisor (GCD) of the numbers of CDs for each type:
- Rock CDs: 14
- Classical CDs: 12
- Pop CDs: 8

#### Step 1: Prime Factorization
- \( 14 = 2 \times 7 \)
- \( 12 = 2^2 \times 3 \)
- \( 8 = 2^3 \)

#### Step 2: Identify the Common Prime Factors
The only common prime factor among 14, 12, and 8 is \( 2 \). The lowest power of 2 that appears in all factorizations is \( 2^1 \).

#### Step 3: Calculate the GCD
\[
\text{GCD} = 2
\]

#### Final Answer:
The greatest number of CDs Oscar can pack in each box is 2.

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Problem 3: Planting Rows

Question: I want to plant 45 sunflower plants, 81 corn plants, and 63 tomato plants in my garden. If I put the same number of plants in each row and each row has only one type of plant, what is the greatest number of plants I can put in one row?

Solution:
To determine the greatest number of plants that can be placed in each row, we need to find the Greatest Common Divisor (GCD) of the numbers of plants for each type:
- Sunflower plants: 45
- Corn plants: 81
- Tomato plants: 63

#### Step 1: Prime Factorization
- \( 45 = 3^2 \times 5 \)
- \( 81 = 3^4 \)
- \( 63 = 3^2 \times 7 \)

#### Step 2: Identify the Common Prime Factors
The only common prime factor among 45, 81, and 63 is \( 3 \). The lowest power of 3 that appears in all factorizations is \( 3^2 \).

#### Step 3: Calculate the GCD
\[
\text{GCD} = 3^2 = 9
\]

#### Final Answer:
The greatest number of plants that can be put in one row is 9.

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Final Answers:

1. 48 days
2. 2 CDs
3. 9 plants

\[
\boxed{48, 2, 9}
\]