1.5 Lowest Common Multiple (LCM)

The lowest common multiple (LCM) is the smallest number that all of them divide into. Same factorisations, opposite rule: multiply EVERY prime that appears anywhere, each at its largest index. The larger index is what every number needs to fit inside.

LCM rule

\text{LCM} = \text{product of } \textbf{all} \text{ prime factors, each with the } \textbf{highest} \text{ index}

Finding the LCM of two numbers

Find the LCM of $40$ and $84$ .

$40 = 2^3 \times 5$
$84 = 2^2 \times 3 \times 7$
Take the highest index for each prime factor: $2^3$ , $3^1$ , $5^1$ and $7^1$ .
$\text{LCM} = 2^3 \times 3 \times 5 \times 7$
$= 8 \times 3 \times 5 \times 7$
$= 840$

Finding the LCM of three numbers

Find the LCM of $8$ , $15$ and $20$ .

$8 = 2^3$
$15 = 3 \times 5$
$20 = 2^2 \times 5$
Take the highest index for each prime: $2^3$ , $3^1$ and $5^1$ .
$\text{LCM} = 2^3 \times 3 \times 5$
$= 8 \times 3 \times 5$
$= 120$

Try it

A practice question on this appears here. Practise primes hcf lcm.

Practise this

Questions on Primes, HCF and LCM, marked as you go, with the working shown.

Practise Primes, HCF and LCM