GCSE Prime factorisation Practice Questions

Split $12$ into a factor pair

Start a factor tree by breaking $12$ into any two factors.

Break down until all prime

$4$ splits into $2 \times 2$, and $3$ is already prime.

Write in index form

Two $2$s become $2²$, giving $12 = 2² \times 3$.

Count the $2s$

Three $2$s multiplied together is $2³$.

The $3$ appears once

There is a single $3$.

Write in index form

So $2 \times 2 \times 2 \times 3 = 2³ \times 3$.

Work out the first power

$2³ = 2 \times 2 \times 2 = 8$.

Work out the second power

$3² = 3 \times 3 = 9$.

Multiply the results

Now multiply $8$ by $9$.

Work it out

$8 \times 9 = 72$.

So the value is

The value is $72$.

State the answer

So $2³ \times 3² = 72$.

Factorise $96$

Write $96$ as a product of primes.

Factorise $144$

Write $144$ as a product of primes.

HCF rule

Take each shared prime to its lowest power.

Shared primes

Both contain $2$ and $3$.

Lowest powers

Lowest power of 2 is 2⁴, of 3 is 3¹.

Work out the HCF

So the HCF is $48$.

LCM rule

Take each prime to its highest power.

Highest powers

$2⁵$ (from $96$) and $3²$ (from $144$).

Work out the LCM

So the LCM is $288$.

State the answers

So HCF = $48$ and LCM = $288$.

Every number factorises uniquely

Each whole number has exactly one prime factorisation.

Listing all factors is slow

A large number can have very many factors.

And error-prone

It is easy to miss a factor or list one twice.

Prime factorisation is compact

The factorisation records everything with just a few prime powers.

For the HCF, use shared primes

Take each shared prime to its lowest power.

This gives the HCF directly

No listing of factors is needed.

For the LCM, use highest powers

Take each prime to its highest power.

This gives the LCM directly

Again, no long list is needed.

You only handle a few numbers

You just compare exponents, not dozens of factors.

So it scales to large numbers

The method stays quick even when the numbers are large.

Example numbers

Take $84$, for instance.

And another

And $360$.

HCF quickly

Lowest shared powers give HCF = $12$ straight away.

LCM quickly

Highest powers give LCM = $2520$ without listing factors.

State the point

Because prime factorisation is unique and compact, it gives a reliable, fast way to find HCF and LCM even for large numbers.