Factors · Greatest common factor · 4.4

GCF in the real world.

GCF answers “largest equal groups” questions and simplifies fractions. To split 18 pens and 30 pencils into identical packs with none left over, use the GCF (6 packs). To simplify 18/30, divide by the GCF.

01 / THE IDEA

See it work.

Watch the method

GCF in the real world

Largest equal packs of 18 and 30 → GCF = 6.

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gcd⁡(18, 30)\gcd(18,\, 30)
1

List the factors of each

Factors of 18 and of 30.

1,2,3,6,9,18  ∣  1,2,3,5,6,10,15,301, 2, 3, 6, 9, 18\;\mid\; 1, 2, 3, 5, 6, 10, 15, 30
2

The greatest common one

The biggest factor they share is the GCF.

gcd⁡(18,30)=6\gcd(18,30) = 6
02 / YOUR TURN

Practice until it feels automatic.

Your turn

Work it out.

Type a number, e.g. 6.

streak 0
best 0
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— / WHY IT WORKS

The idea underneath.

Biggest equal share

When you want the largest identical groups (or lowest-terms fractions), the GCF is the tool. (Links to the Fractions subject.)

Why it works

18/30 = 3/5 after dividing by 6.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Find the GCF of 6 and 10.

1

List the factors of each

Factors of 6 and of 10. 1, 2, 3, 6 | 1, 2, 5, 10

2

The greatest common one

The biggest factor they share is the GCF. gcd(6,10) = 2

EX 2medium

Find the GCF of 12 and 15.

1

List the factors of each

Factors of 12 and of 15. 1, 2, 3, 4, 6, 12 | 1, 3, 5, 15

2

The greatest common one

The biggest factor they share is the GCF. gcd(12,15) = 3

EX 3hard

Find the GCF of 15 and 21.

1

List the factors of each

Factors of 15 and of 21. 1, 3, 5, 15 | 1, 3, 7, 21

2

The greatest common one

The biggest factor they share is the GCF. gcd(15,21) = 3

04 / COMMON TRAPS

Where students slip.

Trap

Using the LCM for sharing.

Fix

Equal groups with none left → GCF.

Trap

Not dividing the fraction by the full GCF.

Fix

Divide top and bottom by the GCF to simplify in one step.

05 / QUIZ

Test yourself.

The Test

12 problems across three tiers. Auto-graded on submit. Hints and a full walkthrough on every question. Your work is saved on this device — sign in to keep it across devices.

0/12Answered
How to answerType just the value. Formatting won't trip you up — we ignore spaces, capital letters, commas, and subscript style, so t₁ = 5, d = 3 and t1=5,d=3 both count. For a list of numbers, separate them with commas (e.g. 3, 10, 17).
Core

Core (4)

C1

Find the GCF of 6 and 14.

Need a hint?
  • “Largest equal groups” → GCF.
  • Simplify a fraction by its GCF.
C2

Find the GCF of 10 and 10.

Need a hint?
  • “Largest equal groups” → GCF.
  • Simplify a fraction by its GCF.
C3

Find the GCF of 12 and 15.

Need a hint?
  • “Largest equal groups” → GCF.
  • Simplify a fraction by its GCF.
C4

Find the GCF of 12 and 21.

Need a hint?
  • “Largest equal groups” → GCF.
  • Simplify a fraction by its GCF.
Apply

Apply (4)

A1

Find the GCF of 9 and 15.

Need a hint?
  • “Largest equal groups” → GCF.
  • Simplify a fraction by its GCF.
A2

Find the GCF of 16 and 12.

Need a hint?
  • “Largest equal groups” → GCF.
  • Simplify a fraction by its GCF.
A3

Find the GCF of 6 and 14.

Need a hint?
  • “Largest equal groups” → GCF.
  • Simplify a fraction by its GCF.
A4

Find the GCF of 24 and 18.

Need a hint?
  • “Largest equal groups” → GCF.
  • Simplify a fraction by its GCF.
Extension

Extension (4)

E1

Find the GCF of 12 and 15.

Need a hint?
  • “Largest equal groups” → GCF.
  • Simplify a fraction by its GCF.
E2

Find the GCF of 20 and 20.

Need a hint?
  • “Largest equal groups” → GCF.
  • Simplify a fraction by its GCF.
E3

Find the GCF of 15 and 21.

Need a hint?
  • “Largest equal groups” → GCF.
  • Simplify a fraction by its GCF.
E4

Find the GCF of 18 and 18.

Need a hint?
  • “Largest equal groups” → GCF.
  • Simplify a fraction by its GCF.