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What is Prime Factorisation?

Every composite number can be written as a product of prime numbers. This “fingerprint” is unique to every number.

Example: 56 $56 = 8 \times 7$ 7 is prime. 8 is composite ( $2 \times 4$ ). $56 = 2 \times 4 \times 7$ 4 is composite ( $2 \times 2$ ). $56 = 2 \times 2 \times 2 \times 7$ Now, all factors (2, 2, 2, 7) are prime. This is the Prime Factorisation of 56.

Factor Trees

A factor tree is a visual way to find prime factors. Let’s factorise 36.

Result: $36 = 3 \times 3 \times 2 \times 2$ . (The blue nodes are the prime numbers).

Tip

Order doesn’t matter: $2 \times 2 \times 3 \times 3$ is the same as $3 \times 2 \times 3 \times 2$ . However, we usually write them in increasing order: $2 \times 2 \times 3 \times 3$ .

Applications

1. Checking for Co-primes

If two numbers have no common prime factors in their factorisation, they are co-prime.

$40 = 2 \times 2 \times 2 \times 5$
$231 = 3 \times 7 \times 11$
No matching primes. 40 and 231 are co-prime.

2. Checking Divisibility

Is 168 divisible by 12?

$168 = 2 \times 2 \times 2 \times 3 \times 7$
$12 = 2 \times 2 \times 3$
Does 168 contain all the “parts” of 12?
- It has two 2s (Yes).
- It has one 3 (Yes).
Therefore, $168 = (2 \times 2 \times 3) \times (2 \times 7) = 12 \times 14$ .
Yes, 168 is divisible by 12.