Prime Factorisation

Prime factors of a number are the prime numbers that divide it exactly.

1.24
All factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Prime factors: 2 and 3

• F₂₄ = 1, 2, 3, 4, 6, 8, 12, 24
• P.F₂₄ = 2 and 3

2. 50
Factors of 50: 1, 2, 5, 10, 25, 50
Prime factors: 2 and 5

• F₅₀ =  = 1, 2, 3, 4, 6, 8, 12, 24
• P.F₅₀ = 2 and 5

3. 64

Factors of 64: 1, 2, 4, 8, 16, 32, 64
Prime factor: 2

• F₆₄ = 1, 2, 4, 8, 16, 32, 64
• P.F₆₄ = 2

When a number is written as a product of two numbers, we can find its prime factorisation by factoring each part.

Write it as a product of two factors:

72 = 12 × 6

Now factor each:

• 12 = 2 × 2 × 3

• 6 = 2 × 3

The prime factors of a product are simply the combined prime factors of its factors.

Two numbers are coprime if they have no common prime factors.

Example 1: 56 and 63

Prime factorisation:

• 56 = 2 × 2 × 2 × 7

• 63 = 3 × 3 × 7

Common prime factor = 7
Not coprime

Example 2: 242 and 195

Prime factorisation:

• 242 = 2 × 11 × 11

• 195 = 3 × 5 × 13

No common prime factors
242 and 195 are coprime.

Rule: If prime factors of B are included in the prime factors of A → A is divisible by B.

Example 1: Is 168 divisible by 12?

Prime factorisation:

• 168 = 2 × 2 × 2 × 3 × 7

• 12 = 2 × 2 × 3

Prime factors of 12 exist inside factors of 168
168 is divisible by 12

Example 2: Is 75 divisible by 21?

Prime factorisation:

• 75 = 3 × 5 × 5

• 21 = 3 × 7

Factor 7 is not present in 75

75 is NOT divisible by 21

• Prime factors are prime numbers that divide a number exactly

• Order doesn't matter when multiplying prime factors

• Two numbers are co-prime if they share no common prime factors

• A number A is divisible by B if all the prime factors of B are present in the prime factorisation of A.