A factor of a number divides into it exactly, with no remainder — factors are always less than or equal to the number. A multiple is what you get by multiplying a number by a whole number — multiples are always greater than or equal to the number (its times table).

Work through pairs systematically: start with $1 \times \text{the number}$, then test $2$, then $3$, and so on, stopping once the pair of factors starts to repeat.

Example: Find the factors of 12.

$1\times12$, $2\times6$, $3\times4$ — testing $4$ would just repeat the pair already found ($4\times3$), so stop here.

The factors of 12 are $1, 2, 3, 4, 6, 12$.

Multiply the number by $1, 2, 3, \dots$ in turn — this is just its times table.

Example: The first five multiples of 8: $8, 16, 24, 32, 40$.

A prime number has exactly two factors: 1 and itself. The number 1 is not prime (it only has one factor), and 2 is the only even prime (every other even number is also divisible by 2, giving it a third factor).

List the factors of each number, identify the factors they share, and pick the largest one.

Example: Find the HCF of 18 and 24.

Factors of 18: $1,2,3,6,9,18$. Factors of 24: $1,2,3,4,6,8,12,24$. Shared factors: $1,2,3,6$. The HCF is $\mathbf{6}$.

List the multiples of each number until you spot the first one that appears in both lists.

Example: Find the LCM of 4 and 5.

Multiples of 4: $4,8,12,16,\mathbf{20},24\dots$. Multiples of 5: $5,10,15,\mathbf{20},25\dots$. The first match is $\mathbf{20}$, so the LCM is $20$.

Two bus routes both stop at the same station: Route A every 15 minutes, Route B every 20 minutes. If both buses leave the station together at 9:00 a.m., when is the next time they both leave together?

1. This asks for the LCM of 15 and 20 (the time after which both cycles realign).
2. Multiples of 15: $15, 30, 45, \mathbf{60}, 75\dots$. Multiples of 20: $20, 40, \mathbf{60}, 80\dots$.
3. The first shared multiple is $60$, so the LCM is $60$ minutes.
4. $9{:}00$ a.m. $+ 60$ minutes $= 10{:}00$ a.m.