Every whole number bigger than 1 can be written as a product of prime numbers in exactly one way. Split the number into any factor pair, then keep splitting each branch until every end is prime. Collect all the primes at the ends, including repeats, and write them smallest first: 60 = 2 × 2 × 3 × 5. Index notation groups repeats: 2 × 2 × 2 is 2³, which means 2 multiplied by itself three times.
Explore this idea with maths tools
Maths you can move
Build 8 + 5
Tap two gold counters to fill the ten-frame.
8 in the frame + 5 waiting. The total stays the same.
Words to know
- prime factor
- a factor that is a prime number
- factor tree
- a diagram that splits a number into factors until all are prime
- index
- the small raised number telling how many times to multiply
Worked examples
Write 60 as a product of prime factors, smallest first.
- 1.60 = 6 × 10, 6 = 2 × 3, 10 = 2 × 5.
- 2.The primes at the ends are 2, 3, 2 and 5.
= 2 × 2 × 3 × 5
Write 126 as a product of prime factors, smallest first.
- 1.126 ÷ 2 = 63, 63 ÷ 3 = 21, 21 ÷ 3 = 7.
- 2.7 is prime, so stop.
= 2 × 3 × 3 × 7
What number is 2³ × 5?
- 1.2³ = 2 × 2 × 2 = 8.
- 2.8 × 5 = 40.
= 40
For parents
Method: Factor tree
What to say: Is every number at the end of a branch prime? Did you include the repeats?
Common mistakes:
- Stops splitting too early and leaves a composite number in the product.
- Writes each prime only once, even when it divides the number more than once.
- Includes 1 in the product of primes, thinking 1 is prime.
- Reads a power as multiplication by the index (2³ as 2 × 3).
- Ignores the powers and multiplies only the bases.
- Works out each power correctly, then adds them instead of multiplying.