Prime factorisation writes a natural number as a product of primes, numbers that cannot be split any further.
Keep dividing a composite number, like taking a model apart block by block, and only primes are left; their product is fixed, apart from the order you multiply in.
Group repeated prime factors as powers, then add 1 to each exponent and multiply, and you know at once how many divisors the number has.
On this page you change the number to see its divisor grid and factor tree, then pick out the GCD and LCM from an exponent table for two numbers.
Divisors, primes and prime factors
To count every divisor of a number, or to find the GCD and LCM of two numbers, it is quicker to first rewrite the number as a product of primes that cannot be split further. This rewriting is called prime factorisation, and repeated prime factors are grouped as powers. What makes the method trustworthy is that whatever order you divide in, you always end up with the same result. The divisor grid below starts by showing where the numbers that divide the chosen number are, and which of them are prime.
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Divisor grid: gold circles are prime factors, blue circles the other divisors
36
The divisor grid has one cell for each number from 1 up to the chosen number (at most 48). Numbers that divide the chosen number exactly are coloured blue, and those among them that are prime are gold: the gold circles are the prime factors. Choose 36, for example, and 1, 2, 3, 4, 6, 9, 12, 18 and 36 are coloured, with only 2 and 3 in gold. Choose a prime and only two cells are coloured: 1 and the number itself.
🧱 Divisors, primes and prime factors
①A divisor of a number divides it exactly. The divisors of 12 are 1, 2, 3, 4, 6 and 12
②A prime is a natural number greater than 1 whose only divisors are 1 and itself: 2, 3, 5, 7, 11, 13, …
③A natural number greater than 1 that is not prime is composite. 1 is neither prime nor composite
④A divisor that is prime is a prime factor. The prime factors of 12 are 2 and 3
The factor tree
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Factor tree: split off the smallest prime factor each time until only primes are left as leaves
36
How to factorise
Divide by the smallest prime, 2, first and work upwards
2 → 3 → 5 → 7 → … repeat until the quotient is 1
Writing with powers
72 = 23 × 32
Group repeated prime factors as powers
📝 Writing the division as a ladder
①Write each prime you divide by on the left and the quotient underneath
②For 72: 72 ÷ 2 = 36, 36 ÷ 2 = 18, 18 ÷ 2 = 9, 9 ÷ 3 = 3, stopping when the quotient is the prime 3
③Multiply the primes you divided by and the last quotient: 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²
④In the tree picture, the numbers dropping to the left leaves are the same as the numbers on the left of the ladder
Counting divisors
It is easy to think of 1 as prime, or to think that the order of dividing changes the factorisation. 1 has only one divisor, itself, so it is neither prime nor composite, and 2 is the only even prime. Whether you start dividing 72 by 2 or by 3, you get 2³ × 3² either way. Keep dividing until the quotient is prime; stop early and a composite number is left behind.
①Every divisor of 72 = 2³ × 3² is made by multiplying some number of 2s and some number of 3s
②You can use 2 zero, one, two or three times: 4 choices
③You can use 3 zero, one or two times: 3 choices
④Pairing the choices gives 4 × 3 = 12, so there are 12 divisors. The key is not to forget the "zero times" option
Linking to GCD and LCM
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Picking the GCD and LCM from the exponent table of two numbers
12
18
GCD
Multiply the shared prime factors, each with the smaller exponent
Use only prime factors that both numbers have
LCM
Multiply every prime factor, each with the larger exponent
Use every prime factor that appears in either number
🔍 GCD × LCM = A × B
①The GCD times the LCM equals the product of the two numbers
②In the exponent table, the smaller plus the larger exponent in each column equals the sum of the two numbers' exponents
③So once you know the GCD, the LCM is A × B ÷ GCD
Check with examples
A prime factorisation problem is solved by dividing by the smallest prime first and repeating until the quotient is prime. Group repeated primes as powers, and if you are asked for the number of divisors, add 1 to each exponent and multiply. Starting with a larger prime gives the same result, but checking divisibility becomes more awkward. The two examples below follow exactly this order.
Example 1
Write 60 as a product of prime factors.
1
Divide by the smallest prime, 2: 60 ÷ 2 = 30, then 30 ÷ 2 = 15.
60 = 2 × 2 × 15
2
15 = 3 × 5, so tidy up with powers.
60 = 2² × 3 × 5
▸ 60 = 2² × 3 × 5
Choose 60 in the tree picture and 2, 2 and 3 drop off to the left in turn, leaving 5 at the end.
Example 2
How many divisors does 72 have?
1
First factorise.
72 = 2³ × 3²
2
Add 1 to each exponent and multiply.
(3 + 1)(2 + 1) = 4 × 3 = 12
▸ 12
Choose 72 in the tree picture and "(3 + 1) × (2 + 1) = 12" appears underneath.
Exam checklist
Once numbers are factorised, the GCD and LCM in the next chapter can be picked out straight away. The GCD multiplies the shared prime factors, each with the smaller exponent, and the LCM multiplies every prime factor, each with the larger exponent. Swapping the two rules is a common slip, so write the exponents of the two numbers in rows side by side, as in the table above, and compare them column by column.
The shape of a prime factorisation
N = p1a_1 × p2a_2 × ...
Every natural number greater than 1 can be written as a product of primes, in only one way apart from order
Grade 7 school test style
60 is multiplied by a natural number to make a perfect square. What is the smallest number it can be multiplied by?
①3
②5
③15
④30
⑤60
▸ ③ 15
1
A perfect square has only even exponents in its prime factorisation.
60 = 2² × 3 × 5
2
Multiply by one more 3 and one more 5 to make the odd exponents even.
60 × (3 × 5) = 900 = 30², multiplier = 15
🎯 Exam checklist
①1 is neither prime nor composite
②The smallest prime is 2, and it is the only even prime
③A prime factorisation is unique apart from the order of the factors
④The number of divisors is the product of (exponent + 1)