Divisibility, prime factorisation, GCF and LCM

Divisibility, prime factorisation, GCF and LCM

Divisibility, prime factorisation, GCF and LCM

In Year 4 you met factors, multiples and prime numbers. Year 5 turns those ideas into reliable tools: shortcuts for spotting divisibility, a clean way to break any number into primes, and two new "common factor / common multiple" numbers you'll meet again and again — especially when you work with fractions.

Divisibility rules — shortcuts you can trust

You don't have to actually divide to know whether one number divides another. Each small divisor has a rule:

DivisorThe rule
2Last digit is even (0, 2, 4, 6, 8).
5Ends in 0 or 5.
10Ends in 0.
3The digits add up to a multiple of 3.
9The digits add up to a multiple of 9.
4The last two digits form a number divisible by 4.
6Divisible by both 2 and 3.

Example — is 246 divisible by 3? Digit sum 2 + 4 + 6 = 12. 12 is in the 3 times table → yes.

Example — is 246 divisible by 4? Last two digits are 46. 46 ÷ 4 = 11 r 2 → no.

Prime factorisation — the "DNA" of a number

Every whole number bigger than 1 can be written as a product of primes in exactly one way (ignoring the order). That's its prime factorisation.

How to do it:

  1. Start with the number. Try dividing by the smallest prime that fits — 2 if it's even, otherwise 3, then 5, 7, …
  2. Write the prime factor down. Replace the number with the quotient.
  3. Keep going until the quotient is 1.

Example — 60:

60 ÷ 2 = 30, 30 ÷ 2 = 15, 15 ÷ 3 = 5, 5 ÷ 5 = 1

60 = 2 × 2 × 3 × 5

A handy way to write the steps is a factor tree:

Greatest common factor (GCF)

The greatest common factor of two numbers is the largest number that divides both. Two ways to find it:

Method 1 — common primes. Prime-factorise both. Multiply the primes they share.

12 = 2 × 2 × 3

18 = 2 × 3 × 3

Shared: one 2 and one 3 → GCF = 2 × 3 = 6

Method 2 — keep dividing. Pick any common factor you spot, divide both sides, repeat.

12, 18 → both ÷ 2 → 6, 9 → both ÷ 3 → 2, 3 → no common factor left

Multiply what you divided out: 2 × 3 = 6

Least common multiple (LCM)

The least common multiple is the smallest number that both numbers divide into.

Quick formula: LCM(a, b) = (a × b) ÷ GCF(a, b).

LCM(12, 18) = (12 × 18) ÷ 6 = 216 ÷ 6 = 36

Or list multiples of the bigger number until one is divisible by the smaller:

Multiples of 18: 18, 36 → 36 ÷ 12 = 3 (yes) → LCM = 36

LCM is exactly what you need when adding fractions with different denominators — the common denominator that keeps the numbers smallest.

Practice