2.2 Prime factorisation, HCF and LCM — Learn

Prerequisites: factor pairs, multiplication facts and prime numbers.

Objectives: express an integer as a product of primes; find the highest common factor (HCF) and lowest common multiple (LCM).

Covers: N4.

Prime factorisation

A prime factorisation writes a number as a product of prime numbers. Divide by small primes repeatedly until every factor is prime.

Worked example. Write 60 as a product of prime factors.

$60=2\times30$ because 2 is a prime factor.

$30=2\times15$, so $60=2\times2\times15$.

$15=3\times5$, so $60=2^2\times3\times5$.

The index 2 means that the prime factor 2 appears twice.

HCF and LCM

The highest common factor is the greatest factor shared by two numbers. The lowest common multiple is the smallest positive multiple shared by them.

Use prime factors for 24 and 36:

For the HCF, take each shared prime with the smaller power: $2^2\times3=12$.

For the LCM, take every prime with the larger power: $2^3\times3^2=72$.

Worked example. A club packs 24 blue and 36 red counters equally, using all counters. Find the greatest possible number of identical packs.

The pack count must divide both 24 and 36.

Their HCF is $12$, so the greatest possible count is 12 packs.

Each pack then has $24\div12=2$ blue and $36\div12=3$ red counters.

Watch out. You may use the LCM when a problem asks for the greatest equal group size. A group size must divide each total, so use common factors and choose the greatest.

Progress check

The same counters can be arranged in equal repeating rows. The first shared row length is the LCM, 72; explain why it is larger than either starting number.

You should now be able to

• write a number as a product of prime factors;
• find an HCF using the smaller shared prime powers;
• find an LCM using the larger prime powers;
• explain whether a context needs an HCF or an LCM.