Prime factorisation

Breaking a number into primes

Take 56. It is composite: 56 = 4 × 14. Now 14 is also composite, 14 = 2 × 7, so 56 = 4 × 2 × 7. And 4 = 2 × 2, so 56 = 2 × 2 × 2 × 7. Now every factor is prime and cannot be broken further. Writing a number as a product of primes is its prime factorisation, and the primes in it, here 2 and 7, are its prime factors.

• Every number greater than 1 has a prime factorisation.
• The number 1 has none, because no prime divides it.
• A prime is its own prime factorisation: 7 = 7.

Factor trees

A factor tree keeps splitting each composite number into two factors until only primes are left.

Whether you start 36 as 2 × 18, 3 × 12, 4 × 9 or 6 × 6, you always end with two 2s and two 3s. It is a remarkable fact that every number has only one prime factorisation, apart from the order of the factors.

Does the order matter?

No. Think of 30 cubes stacked as a box 2 cubes high, 3 cubes wide and 5 cubes long: counting them layer by layer in any direction gives 30, so 2 × 3 × 5 = 3 × 5 × 2 = 5 × 2 × 3. We can multiply numbers in any order and get the same result, which you will later study as commutativity and associativity. By habit we write the primes in increasing order: 225 = 3 × 3 × 5 × 5.

Factorising a product

To factorise a product, factorise each part and put them together. Since 72 = 12 × 6, 12 = 2 × 2 × 3 and 6 = 2 × 3, we get 72 = 2 × 2 × 3 × 2 × 3 = 2 × 2 × 2 × 3 × 3. Each prime appears in 72 as many times as in 12 and 6 together. In the same way, 108 × 75 = (2 × 2 × 3 × 3 × 3) × (3 × 5 × 5) = 2 × 2 × 3 × 3 × 3 × 3 × 5 × 5.

Worked examples

Number	Prime factorisation
64	2 × 2 × 2 × 2 × 2 × 2
104	2 × 2 × 2 × 13
105	3 × 5 × 7
243	3 × 3 × 3 × 3 × 3
1331	11 × 11 × 11
1000	2 × 2 × 2 × 5 × 5 × 5

The smallest number with three different prime factors is 2 × 3 × 5 = 30, and with four it is 2 × 3 × 5 × 7 = 210.

Using prime factorisation

To check co-primes: two numbers are co-prime if no prime appears in both factorisations. 80 = 2 × 2 × 2 × 2 × 5 and 63 = 3 × 3 × 7 share none, so they are co-prime.

To check divisibility: one number divides another if its prime factorisation is included in the other’s.

• Is 168 divisible by 12? 168 = 2 × 2 × 2 × 3 × 7 contains 12 = 2 × 2 × 3, so yes: 168 = 12 × 14.
• Is 75 divisible by 21? 21 = 3 × 7, but 7 is not a prime factor of 75 = 3 × 5 × 5, so no.
• Is 42 divisible by 12? 12 needs two 2s, but 42 = 2 × 3 × 7 has only one, so no.

For checking whether two numbers share a factor, read co-prime numbers; for quick tests, read divisibility rules. For the textbook’s questions, see the Chapter 5 NCERT solutions; for the whole chapter, see the Chapter 5 notes.