GCF and LCM Worksheet- Common Core Practice Problems
What Is GCF and LCM, and Why Do You Need Both?
GCF stands for Greatest Common Factor. LCM stands for Least Common Multiple. These are two of the most tested skills in middle school math, and they show up on standardized tests, homework, and real-world problems alike.
Most students mix them up. That's not surprising — they sound similar and both involve finding relationships between numbers. The difference matters, though. Getting them confused will cost you points.
GCF answers: "What is the biggest number that divides evenly into both numbers?"
LCM answers: "What is the smallest number that both numbers divide into evenly?"
That single distinction is the whole game. Master it, and every GCF and LCM problem becomes straightforward.
How to Find GCF: Two Methods That Actually Work
Method 1: List All Factors
Write out every factor for each number, then find the biggest one they share.
Example: Find GCF of 24 and 36
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Common factors: 1, 2, 3, 4, 6, 12
GCF = 12
Method 2: Prime Factorization
Break each number into its prime factors, then multiply the common primes.
Example: Find GCF of 48 and 180
48 = 2 × 2 × 2 × 2 × 3
180 = 2 × 2 × 3 × 3 × 5
Common primes: 2 × 2 × 3 = 12
GCF = 12
Prime factorization is faster for large numbers. Listing factors is easier to understand. Use whichever clicks for you.
How to Find LCM: Two Methods That Actually Work
Method 1: List Multiples
Write out multiples of each number until you find the smallest one they share.
Example: Find LCM of 6 and 8
Multiples of 6: 6, 12, 18, 24, 30...
Multiples of 8: 8, 16, 24, 32...
LCM = 24
Method 2: Prime Factorization (All Factors)
Take each prime number that appears in either factorization, use it the most times it appears in any single number.
Example: Find LCM of 12 and 18
12 = 2 × 2 × 3
18 = 2 × 3 × 3
LCM uses: 2 (twice, from 12) × 3 (twice, from 18) = 36
LCM = 36
Giant GCF and LCM Practice Problems
Work through these. No calculator until you've tried by hand.
GCF Problems
- Find the GCF of 45 and 75
- Find the GCF of 84 and 126
- Find the GCF of 144 and 192
- Find the GCF of 56, 84, and 140
- Find the GCF of 99 and 121
LCM Problems
- Find the LCM of 9 and 12
- Find the LCM of 14 and 21
- Find the LCM of 8, 12, and 20
- Find the LCM of 15 and 25
- Find the LCM of 7 and 13
Combined Problems
- Find two numbers with GCF = 8 and LCM = 48
- The GCF of two numbers is 6. One number is 18. What could the other number be?
- A clock chimes every 12 minutes and every 18 minutes. When will it chime together again?
GCF and LCM Word Problems
Most students freeze when they see a story problem. The fix is simple: translate the problem into math before you solve it.
Problem 1: Sarah has 24 roses and 36 lilies. She wants to make identical bouquets with no flowers left over. What is the greatest number of bouquets she can make?
Translation: Find the GCF of 24 and 36.
Answer: 12 bouquets.
Problem 2: Two traffic lights blink every 8 seconds and every 12 seconds. If they blink together now, in how many seconds will they blink together again?
Translation: Find the LCM of 8 and 12.
Answer: 24 seconds.
Problem 3: A baker has 40 chocolate chip cookies and 52 oatmeal cookies. He packages them in identical gift boxes with no cookies left out. What is the maximum number of boxes he can make?
Translation: Find the GCF of 40 and 52.
Answer: 4 boxes.
GCF vs LCM: Side-by-Side Comparison
| Concept | Question It Answers | Best Method | Example |
|---|---|---|---|
| GCF | Biggest number that divides into both | List factors or prime factorization | GCF of 36 and 48 = 12 |
| LCM | Smallest number both divide into | List multiples or prime factorization | LCM of 5 and 7 = 35 |
Common Mistakes That Blow the Answer
- Confusing GCF with LCM. This is the number one error. Read the question twice before you solve.
- Stopping too early on multiples. Students often stop at the first common multiple they find. Make sure it's the least one.
- Missing prime factors. When using prime factorization for LCM, you need the highest power of each prime that appears in any factorization.
- Rushing on word problems. The math is simple. The hard part is identifying which operation the problem is asking for.
Quick Reference Cheat Sheet
- GCF = factor, divide, share → how big can pieces be when split equally
- LCM = multiple, stack, combine → when will things line up again
- Prime factorization works for both — just use different rules
- GCF is always ≤ the smaller number
- LCM is always ≥ the larger number
Getting Started: Your First Practice Set
Grab a pencil and scratch paper. Work these ten problems without checking the answers first.
- GCF of 27 and 45
- LCM of 10 and 15
- GCF of 72 and 96
- LCM of 6, 9, and 12
- GCF of 34 and 85
- LCM of 4 and 13
- GCF of 48, 64, and 80
- LCM of 16 and 24
- Find GCF × LCM of 8 and 14. Compare to 8 × 14.
- Two bells ring every 9 and 12 seconds. When do they ring together?
Check your answers. If you missed any, go back and figure out whether you confused GCF with LCM or made a calculation error. That's the only way to improve.
Why These Skills Keep Showing Up
GCF and LCM are foundational for:
- Adding and subtracting fractions
- Simplifying algebraic expressions
- Solving diophantine equations
- Real scheduling problems
You will see these again in Algebra 1, Geometry, and on the SAT. The time you spend mastering them now pays off later.