Greatest Common Factor (GCF)

Free sample questions, a clear explanation, and 5 practice skills with an AI tutor that guides without giving the answer away.

Concept Review

Greatest Common Factor: Finding the Perfect Team Captain

Imagine you're organizing teams for a school event. You have 24 students from one class and 18 from another. What's the largest team size that divides both groups evenly, with no students left out? This is exactly what the Greatest Common Factor (GCF) helps us discover!

The Greatest Common Factor is the largest number that divides evenly into two or more numbers. To find it, we need to understand factors — numbers that multiply together to make another number.

Finding Factors: The Division Detective Method

Let's solve our team problem step by step. First, we'll list all the factors of each number by testing which numbers divide evenly (no remainders).

Factors of 24:

1 × 24 = 24 ✓ 2 × 12 = 24 ✓ 3 × 8 = 24 ✓ 4 × 6 = 24 ✓

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Factors of 18:

1 × 18 = 18 ✓ 2 × 9 = 18 ✓ 3 × 6 = 18 ✓

Factors of 18: 1, 2, 3, 6, 9, 18

Now we identify the common factors — numbers that appear in both lists: 1, 2, 3, and 6. The largest of these is 6, making it our Greatest Common Factor!

🔑 Key Insight

The GCF is never larger than the smaller number you're comparing. If you're finding the GCF of 15 and 40, it can't be bigger than 15. This saves you time — you only need to check factors up to the smaller number!

Real-World GCF in Action

Back to our teams: with a GCF of 6, we can make teams of 6 students each. The 24 students form 4 teams, and the 18 students form 3 teams — perfect! No other team size larger than 6 would work for both groups.

You'll find GCF everywhere: organizing equal rows of desks, dividing pizza slices fairly among friends, or finding the largest tile size that fits perfectly along two different wall lengths.

🎯 Key Takeaway

The Greatest Common Factor isn't just about numbers on paper — it's about finding the perfect "size" that works for multiple situations. Whether you're the team captain organizing groups or an architect planning tile layouts, the GCF helps you find the largest solution that fits everything perfectly.

Sample questions

1. What is the GCF of 12 and 18?

○ 6

○ 3

○ 36

✓ 6? Factors of 12: 1,2,3,4,6,12; of 18: 1,2,3,6,9,18; common: 1,2,3,6; greatest is 6

○ 6

Answer: 6? Factors of 12: 1,2,3,4,6,12; of 18: 1,2,3,6,9,18; common: 1,2,3,6; greatest is 6 — The largest common factor is 6.

2. Find the GCF of 24 and 36.

○ 12

○ 6

○ 72

✓ 12? Factors of 24: 1,2,3,4,6,8,12,24; of 36: 1,2,3,4,6,9,12,18,36; greatest common is 12

○ 12

Answer: 12? Factors of 24: 1,2,3,4,6,8,12,24; of 36: 1,2,3,4,6,9,12,18,36; greatest common is 12 — The GCF is 12.

3. What is the GCF of 15 and 28?

○ 1

○ 7

○ 5

✓ 1? Factors of 15: 1,3,5,15; of 28: 1,2,4,7,14,28; only common factor is 1

○ 1

Answer: 1? Factors of 15: 1,3,5,15; of 28: 1,2,4,7,14,28; only common factor is 1 — They are relatively prime, so GCF = 1.

Skills in this topic

List factors to find the greatest common factor of two whole numbers less than or equal to 100
Use prime factorization trees to find the GCF
Use the Distributive Property to express a sum of two whole numbers (e.g., 36 + 8 as 4(9 + 2))
Solve real-world problems involving grouping identical items using GCF
Identify when a word problem requires finding the GCF

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