GCF and LCM Word Problems | Worsheets library - Free Printable

Let's solve each of the three word problems step by step, focusing on Greatest Common Factor (GCF) and Least Common Multiple (LCM) concepts.

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Problem 1: Hilary's Activities

> At the gym, Hilary 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?

Step-by-step Solution:

We are looking for the first day when all three activities coincide again. This is a classic LCM (Least Common Multiple) problem.

- Swimming: every 6 days
- Running: every 4 days
- Cycling: every 16 days

We need to find the LCM of 6, 4, and 16.

#### Step 1: Prime factorization
- 6 = 2 × 3
- 4 = 2²
- 16 = 2⁴

#### Step 2: Take the highest power of each prime
- 2⁴ (from 16)
- 3¹ (from 6)

So, LCM = 2⁴ × 3 = 16 × 3 = 48

✔ Answer: Hilary will do all three activities again on the same day in 48 days.

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Problem 2: Oscar’s CDs

> 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?

Step-by-step Solution:

We want to pack equal numbers of CDs in each box, with only one type per box, and we want the greatest possible number per box.

This means we're looking for the largest number that divides all three quantities evenly — this is the GCF (Greatest Common Factor) of 14, 12, and 8.

#### Step 1: Find GCF of 14, 12, and 8

- Factors of 14: 1, 2, 7, 14
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 8: 1, 2, 4, 8

Common factors: 1, 2

The greatest common factor is 2.

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

(He would need:
- 14 ÷ 2 = 7 boxes for rock
- 12 ÷ 2 = 6 boxes for classical
- 8 ÷ 2 = 4 boxes for pop)

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Problem 3: Planting in Rows

> 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’s the greatest number of plants I can put in one row?

Step-by-step Solution:

Again, we want the same number of plants per row, only one type per row, and we want the maximum number per row.

So, we need the GCF of 45, 81, and 63.

#### Step 1: Prime factorization

- 45 = 3² × 5
- 81 = 3⁴
- 63 = 3² × 7

Now take the lowest power of common primes:

- Only 3 is common to all.
- Lowest power: 3² = 9

So, GCF = 9

✔ Answer: The greatest number of plants you can put in one row is 9.

(You’d have:
- 45 ÷ 9 = 5 rows of sunflowers
- 81 ÷ 9 = 9 rows of corn
- 63 ÷ 9 = 7 rows of tomatoes)

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✔ Final Answers:

1. 48 days
2. 2 CDs per box
3. 9 plants per row

These solutions use LCM for overlapping events and GCF for equal grouping/division.