GCF and LCM Word Problems #1 worksheet with four math word problems involving real-life scenarios.

Let’s solve each problem one by one, step by step.

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Problem 1: GCF and LCM Word Problems #1

> At the gym, Hillary swims every 6 days, runs every 4 days and cycles every 15 days. If she did all three activities today, in how many days will she do all three activities on the same day again?

We need to find when all three activities line up again — that means we’re looking for the Least Common Multiple (LCM) of 6, 4, and 15.

Step 1: Prime factorization
- 6 = 2 × 3
- 4 = 2²
- 15 = 3 × 5

Step 2: Take the highest power of each prime
- 2² (from 4)
- 3¹ (from 6 or 15)
- 5¹ (from 15)

Multiply them:
2² × 3 × 5 = 4 × 3 × 5 = 60

✔ So, Hillary will do all three activities together again in 60 days.

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Problem 2:

> 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?

This is asking for the Greatest Common Factor (GCF) of 14, 12, and 8 — because we want the largest number that divides evenly into all three.

Step 1: List factors
- 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 → Greatest is 2

✔ So, Oscar can pack 2 CDs per box.

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Problem 3:

> 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 if each row has only one type of plant, what is the greatest number of plants I can put in one row?

Again, this is a GCF problem — we want the largest number that divides 45, 81, and 63 evenly.

Step 1: Prime factorization
- 45 = 3² × 5
- 81 = 3⁴
- 63 = 3² × 7

Common prime factor: 3² = 9

✔ So, the greatest number of plants per row is 9.

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Problem 4:

> Cups are sold to a package and plates are sold to a package. If you want to have the same number of each item for a party, what is the least number of packages of each you need to buy?

Wait — the problem doesn’t say how many cups or plates come in a package! That’s missing info.

But since this is from a worksheet titled “GCF and LCM”, and it’s paired with other problems, maybe there was supposed to be numbers like “cups come in packs of 8, plates in packs of 12” — but they’re not here.

Looking back at the image text:
“Cups are sold to a package and plates are sold to a package.” — no quantities given.

Hmm… This seems incomplete. But perhaps in the original context, standard numbers were assumed? Or maybe it’s a typo?

Actually — let me check the image again carefully.

In the user-uploaded image, Problem 4 says:

> Cups are sold to a package and plates are sold to a package. If you want to have the same number of each item for a party, what is the least number of packages of each you need to buy?

Still no numbers. That’s odd.

Wait — maybe it’s implied from previous problems? No.

Alternatively — perhaps this is meant to be solved with variables? But that doesn’t make sense for middle school level.

Hold on — maybe I misread. Let me look again.

Actually, in some versions of this worksheet, Problem 4 usually says something like:

> Cups are sold in packages of 8, plates in packages of 12...

Since that’s common, and without those numbers we can’t solve it, I’ll assume that’s what was intended — otherwise the problem is unsolvable.

So let’s assume:
- Cups come in packs of 8
- Plates come in packs of 12

Then we need the LCM of 8 and 12 to find the smallest number where both match.

Prime factors:
- 8 = 2³
- 12 = 2² × 3

LCM = 2³ × 3 = 8 × 3 = 24

So you need 24 cups and 24 plates.

Number of cup packages: 24 ÷ 8 = 3
Number of plate packages: 24 ÷ 12 = 2

✔ So, least number of packages: 3 packages of cups, 2 packages of plates

*(Note: Since the original problem didn’t specify package sizes, this is an assumption based on typical textbook problems. If your actual worksheet had different numbers, plug those in instead.)*

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Problem 5:

> A full moon occurs every 30 days. If the last full moon occurred on a Friday, how many days will pass before a full moon occurs again on a Friday?

We need the next time a full moon lands on a Friday.

Full moons happen every 30 days. Days of the week repeat every 7 days.

So we need to find the smallest number of days that is a multiple of both 30 (full moon cycle) and 7 (week cycle) — so that the day of the week lines up again.

That’s the LCM of 30 and 7.

30 = 2 × 3 × 5
7 = 7 (prime)

LCM = 2 × 3 × 5 × 7 = 210

✔ So, 210 days will pass before a full moon occurs again on a Friday.

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Final Answer:

1. 60 days
2. 2 CDs per box
3. 9 plants per row
4. 3 packages of cups, 2 packages of plates *(assuming 8 cups/pack, 12 plates/pack)*
5. 210 days