GCF and LCM Word Problems worksheet with five real-world math scenarios.

Let’s solve each problem one by one, using GCF (Greatest Common Factor) or LCM (Least Common Multiple) as appropriate.

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1. Pencils and Erasers — Find the smallest number of packages so that there is exactly 1 eraser per pencil

We are told:
- Pencils come in packages of 12
- Erasers come in packages of 10
- We want equal numbers of pencils and erasers → so we need the Least Common Multiple (LCM) of 12 and 10.

#### Step-by-step:

Find LCM(12, 10):

Prime factorization:
- 12 = 2² × 3
- 10 = 2 × 5

LCM = highest power of all primes: 2² × 3 × 5 = 60

So, we need 60 pencils and 60 erasers.

Now, how many packages?

- Pencil packages: 60 ÷ 12 = 5 packages
- Eraser packages: 60 ÷ 10 = 6 packages

✔ Answer: B. 5 packages of pencils and 6 packages of erasers

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2. Cookies — Divide into identical containers with same number of each kind

We have:
- 30 oatmeal cookies
- 48 chocolate chip cookies

We want to divide them into identical containers, each with the same number of each kind. So we need the Greatest Common Factor (GCF) of 30 and 48 — this will tell us the maximum number of containers we can make, and each container will get:

- 30 ÷ GCF oatmeal cookies
- 48 ÷ GCF chocolate chip cookies

#### Step-by-step:

Find GCF(30, 48)

Prime factorization:
- 30 = 2 × 3 × 5
- 48 = 2⁴ × 3

GCF = common primes with lowest powers: 2 × 3 = 6

So, maximum number of containers = 6

Each container gets:
- 30 ÷ 6 = 5 oatmeal cookies
- 48 ÷ 6 = 8 chocolate chip cookies

✔ Answer: 6 containers, each with 5 oatmeal and 8 chocolate chip cookies

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3. Boxes stacked — Find shortest height where two stacks are equal

Boxes are 12 inches tall and 18 inches tall.

We stack them separately and want the shortest height at which both stacks are equal.

This means we need the LCM of 12 and 18 — the smallest height that is a multiple of both.

#### Step-by-step:

Find LCM(12, 18)

Prime factorization:
- 12 = 2² × 3
- 18 = 2 × 3²

LCM = 2² × 3² = 4 × 9 = 36

✔ Answer: 36 inches

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4. Tours leaving at the same time — Find when they leave together again

- National Capitol tour leaves every 15 minutes
- White House tour leaves every 20 minutes
- Both leave at 8:30 AM

We want the next time they leave together → LCM of 15 and 20

#### Step-by-step:

LCM(15, 20)

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

LCM = 2² × 3 × 5 = 4 × 3 × 5 = 60 minutes

So, they leave together again 60 minutes after 8:30 AM → 9:30 AM

✔ Answer: D. Every 60 minutes

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5. Lights blinking — When do they blink together?

- Light A blinks every 4 minutes
- Light B blinks every 6 minutes
- They start blinking together at time zero
- Question: In 60 seconds, how many times do they blink at the same time?

Wait — 60 seconds is 1 minute, but the lights blink every 4 minutes and 6 minutes — that’s way longer than 1 minute!

That doesn’t make sense unless there’s a typo.

Let’s re-read:

> “Two neon lights are turned on at the same time. One blinks every 4 seconds and the other blinks every 6 seconds. In 60 seconds, how many times will they blink at the same time?”

Ah! Probably meant seconds, not minutes — otherwise, in 60 seconds (1 minute), neither would blink even once if they blinked every 4 and 6 *minutes*.

Assuming it’s seconds (as it makes sense contextually), let’s proceed.

So:
- Light A blinks every 4 seconds
- Light B blinks every 6 seconds
- Both start at time 0
- We want how many times they blink together in 60 seconds

→ This is LCM of 4 and 6

#### Step-by-step:

LCM(4, 6)

4 = 2²
6 = 2 × 3
LCM = 2² × 3 = 12

So they blink together every 12 seconds

Now, from time 0 to 60 seconds, inclusive, how many multiples of 12?

Times: 0, 12, 24, 36, 48, 60 → that’s 6 times

✔ Answer: 6 times

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## ✔ Final Answers:

1. B. 5 packages of pencils and 6 packages of erasers
2. 6 containers, each with 5 oatmeal and 8 chocolate chip cookies
3. 36 inches
4. D. Every 60 minutes
5. 6 times

Let me know if you’d like visual diagrams or further explanation for any part!