Procedure

LCM and HCF of several numbers from prime powers

Arithmetic • Number System • LCM and HCF

Statement

Factorise every number into primes. The LCM is the product of each prime raised to the HIGHEST power in which it appears in any of them; the HCF is the product of each prime raised to the LOWEST power in which it appears in ALL of them, a prime missing from any one number being omitted entirely.

How to use it

1Works for any count of numbers, unlike the two-number identity $HCF \times LCM = ab$ .
2Lay the factorisations in a column with one prime per position, so the highest and lowest powers can be read off by eye.
3A prime that appears in only one of the numbers still enters the LCM, but never the HCF.
4Events that repeat every $a$ , $b$ and $c$ units coincide again after $LCM (a, b, c)$ units — this is the standard route for bells, lights, or runners returning together to a starting point.

Worked example

Find the LCM and the HCF of

12

18

and

30

Step-by-step

1
$12 = 2^{2} \cdot 3, 18 = 2 \cdot 3^{2}, 30 = 2 \cdot 3 \cdot 5$
Why: Factorise each number into primes.
2
$LCM = 2^{2} \cdot 3^{2} \cdot 5 = 180$
Why: Highest power of each prime seen anywhere: $2^{2}$ from $12$ , $3^{2}$ from $18$ , $5$ from $30$ .
3
$HCF = 2 \cdot 3 = 6$
Why: Lowest power present in all three; $5$ is dropped because $12$ and $18$ do not contain it.

Answer

LCM = 180

HCF = 6

Watch out

Taking the highest power for the HCF, or the lowest for the LCM, is the usual slip. The pair identity

HCF \times LCM = ab

does not extend to three or more numbers, so do not use it as a shortcut here.

Builds on

Prime factorisation

Also known as

lcm by prime factorisation, hcf by prime factorisation, highest power rule, lcm of three numbers, gcd of several numbers

Appears in past papers

2020 (I)

Q47