Factors · Greatest common factor · 4.3

GCF by primes.

Using prime factorization: GCF takes the LOWEST power of each prime the numbers share. 24 = 2³×3, 36 = 2²×3². Shared primes: 2 (lowest power 2²) and 3 (lowest power 3¹) → 4×3 = 12.

01 / THE IDEA

See it work.

Factor tree

Split until every leaf is prime.

The prime leaves are the prime factorization.

242122623

24 = 2³ × 3

Watch the method

GCF by primes

24 = 2³×3, 36 = 2²×3² → GCF = 2²×3 = 12.

1 / 2
gcd⁡(24, 36)\gcd(24,\, 36)
1

Prime factorize both

Find each number’s prime factorization.

24=23×3,  36=22×3224 = 2^{3} \times 3,\; 36 = 2^{2} \times 3^{2}
2

Take the lowest power of shared primes

Multiply the lowest power of each common prime.

gcd⁡=12\gcd = 12
02 / YOUR TURN

Practice until it feels automatic.

Your turn

Work it out.

Type a number, e.g. 6.

streak 0
best 0
… =
— / WHY IT WORKS

The idea underneath.

Lowest shared powers

Factor both into primes; for each prime in both, take the smaller exponent, then multiply.

Why it works

min(2³, 2²) = 2², min(3¹, 3²) = 3¹.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Find the GCF of 12 and 18.

1

Prime factorize both

Find each number’s prime factorization. 12 = 2^2 × 3, 18 = 2 × 3^2

2

Take the lowest power of shared primes

Multiply the lowest power of each common prime. gcd = 6

EX 2medium

Find the GCF of 8 and 10.

1

Prime factorize both

Find each number’s prime factorization. 8 = 2^3, 10 = 2 × 5

2

Take the lowest power of shared primes

Multiply the lowest power of each common prime. gcd = 2

EX 3hard

Find the GCF of 12 and 42.

1

Prime factorize both

Find each number’s prime factorization. 12 = 2^2 × 3, 42 = 2 × 3 × 7

2

Take the lowest power of shared primes

Multiply the lowest power of each common prime. gcd = 6

04 / COMMON TRAPS

Where students slip.

Trap

Taking the highest power.

Fix

GCF uses the LOWEST power (that’s LCM’s job for highest).

Trap

Including a prime not in both.

Fix

Only primes shared by both count.

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 8 and 14.

Need a hint?
  • Prime factorize both.
  • Take the lowest power of each shared prime.
C2

Find the GCF of 10 and 10.

Need a hint?
  • Prime factorize both.
  • Take the lowest power of each shared prime.
C3

Find the GCF of 6 and 10.

Need a hint?
  • Prime factorize both.
  • Take the lowest power of each shared prime.
C4

Find the GCF of 9 and 15.

Need a hint?
  • Prime factorize both.
  • Take the lowest power of each shared prime.
Apply

Apply (4)

A1

Find the GCF of 10 and 6.

Need a hint?
  • Prime factorize both.
  • Take the lowest power of each shared prime.
A2

Find the GCF of 10 and 10.

Need a hint?
  • Prime factorize both.
  • Take the lowest power of each shared prime.
A3

Find the GCF of 4 and 6.

Need a hint?
  • Prime factorize both.
  • Take the lowest power of each shared prime.
A4

Find the GCF of 12 and 9.

Need a hint?
  • Prime factorize both.
  • Take the lowest power of each shared prime.
Extension

Extension (4)

E1

Find the GCF of 6 and 9.

Need a hint?
  • Prime factorize both.
  • Take the lowest power of each shared prime.
E2

Find the GCF of 8 and 28.

Need a hint?
  • Prime factorize both.
  • Take the lowest power of each shared prime.
E3

Find the GCF of 8 and 28.

Need a hint?
  • Prime factorize both.
  • Take the lowest power of each shared prime.
E4

Find the GCF of 30 and 18.

Need a hint?
  • Prime factorize both.
  • Take the lowest power of each shared prime.