Factors · Primes & prime factorization · 3.5

One way only.

No matter how you split a number, its prime factorization is always the same (apart from order). 60 = 2²×3×5 whether you start with 6×10 or 4×15. This is the Fundamental Theorem of Arithmetic.

01 / THE IDEA

See it work.

Factor tree

Split until every leaf is prime.

The prime leaves are the prime factorization.

6023021535

60 = 2² × 3 × 5

Watch the method

One way only

60 = 2²×3×5, however you start.

1 / 2
6060
1

Build a factor tree

Split into factors until every leaf is prime.

2×2×3×52 \times 2 \times 3 \times 5
2

Write the prime factorization

Collect the primes (use exponents for repeats).

60=22×3×560 = 2^{2} \times 3 \times 5
02 / YOUR TURN

Practice until it feels automatic.

Your turn

Work it out.

Type primes, e.g. 2^3·3.

streak 0
best 0
… =
— / WHY IT WORKS

The idea underneath.

Always the same

Different factor trees for the same number give the same primes — uniqueness.

Why it works

6×10 and 4×15 both end at 2²·3·5.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Write the prime factorization of 48.

1

Build a factor tree

Split into factors until every leaf is prime. 2 × 2 × 2 × 2 × 3

2

Write the prime factorization

Collect the primes (use exponents for repeats). 48 = 2^4 × 3

EX 2medium

Write the prime factorization of 80.

1

Build a factor tree

Split into factors until every leaf is prime. 2 × 2 × 2 × 2 × 5

2

Write the prime factorization

Collect the primes (use exponents for repeats). 80 = 2^4 × 5

EX 3hard

Write the prime factorization of 42.

1

Build a factor tree

Split into factors until every leaf is prime. 2 × 3 × 7

2

Write the prime factorization

Collect the primes (use exponents for repeats). 42 = 2 × 3 × 7

04 / COMMON TRAPS

Where students slip.

Trap

Thinking different trees give different answers.

Fix

They always reach the same primes.

Trap

Leaving a composite in the answer.

Fix

Break it all the way down to primes.

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

Write the prime factorization of 30.

Need a hint?
  • Any starting pair works.
  • You’ll reach the same primes.
C2

Write the prime factorization of 24.

Need a hint?
  • Any starting pair works.
  • You’ll reach the same primes.
C3

Write the prime factorization of 48.

Need a hint?
  • Any starting pair works.
  • You’ll reach the same primes.
C4

Write the prime factorization of 42.

Need a hint?
  • Any starting pair works.
  • You’ll reach the same primes.
Apply

Apply (4)

A1

Write the prime factorization of 45.

Need a hint?
  • Any starting pair works.
  • You’ll reach the same primes.
A2

Write the prime factorization of 80.

Need a hint?
  • Any starting pair works.
  • You’ll reach the same primes.
A3

Write the prime factorization of 40.

Need a hint?
  • Any starting pair works.
  • You’ll reach the same primes.
A4

Write the prime factorization of 20.

Need a hint?
  • Any starting pair works.
  • You’ll reach the same primes.
Extension

Extension (4)

E1

Write the prime factorization of 50.

Need a hint?
  • Any starting pair works.
  • You’ll reach the same primes.
E2

Write the prime factorization of 80.

Need a hint?
  • Any starting pair works.
  • You’ll reach the same primes.
E3

Write the prime factorization of 36.

Need a hint?
  • Any starting pair works.
  • You’ll reach the same primes.
E4

Write the prime factorization of 40.

Need a hint?
  • Any starting pair works.
  • You’ll reach the same primes.