GCSE Factors, multiples and primes Practice Questions

Find pairs that multiply to $12$

Factors come in pairs. Look for every pair of whole numbers whose product is $12$.

Each pair gives two factors

Write out both numbers from each pair.

List them in order

Put the factors in increasing order.

Find factor pairs

Look for all pairs of whole numbers multiplying to $30$.

Collect the factors

Take both numbers from each pair.

List in order

Arrange from smallest to largest.

What the HCF means

The HCF is the largest factor two numbers have in common.

The HCF is $1$

Here the biggest shared factor is just $1$.

So the only common factor is $1$

They have no other factor in common.

There is a name for this

Numbers whose only common factor is $1$ are called coprime (or relatively prime).

An example

For instance $8$ and $9$ share no factor except $1$.

State the answer

So the two numbers share no common factor other than $1$.

The number divides by $3$ and by $4$

To share equally between $3$ or $4$ with none left over, the total must divide by both $3$ and $4$.

Smallest such number is the LCM

The smallest number that works is the lowest common multiple.

Multiples of $3$

List the multiples of $3$.

Multiples of $4$

List the multiples of $4$.

First common multiple

$12$ is the smallest number in both lists.

So the LCM is

The lowest common multiple is $12$.

Interpret

So the smallest number of cards is $12$.

Check sharing by $3$

$12$ shared between $3$ people gives $4$ each.

Check sharing by $4$

$12$ shared between $4$ people gives $3$ each.

State the answer

So the smallest number of cards is $12$.

Find the HCF of $8$ and $12$

Start with the highest common factor.

Factors of $8$

List the factors of $8$.

Factors of $12$

List the factors of $12$.

Common factors

These appear in both lists.

HCF

The largest common factor is $4$.

What the HCF means

The HCF is the largest number that divides both $8$ and $12$.

When you use the HCF

Use the HCF for problems like the largest tile or longest equal piece.

Now find the LCM

Next find the lowest common multiple.

Multiples of $8$

List the multiples of $8$.

Multiples of $12$

List the multiples of $12$.

LCM

The smallest common multiple is $24$.

What the LCM means

The LCM is the smallest number that both $8$ and $12$ divide into.

When you use the LCM

Use the LCM for problems like bells ringing or lights flashing together again.

Compare their sizes

The HCF ($4$) is at most each number; the LCM ($24$) is at least each number.

State the difference

So the HCF ($4$) is for splitting into equal groups, and the LCM ($24$) is for events coinciding.