1.4 Highest Common Factor (HCF)

The highest common factor (HCF) of two or more numbers is the largest number dividing exactly into all of them. Write each as a product of primes, then multiply the primes appearing in EVERY factorisation, each at its smallest index. The smaller index is all the numbers can share.

HCF rule

\text{HCF} = \text{product of } \textbf{common} \text{ prime factors, each with the } \textbf{smallest} \text{ index}

Finding the HCF of two numbers

Find the HCF of $84$ and $126$ .

$84 = 2^2 \times 3 \times 7$
$126 = 2 \times 3^2 \times 7$
Common prime factors: $2$ , $3$ and $7$ .
Take the smaller index for each: $2^1$ , $3^1$ and $7^1$ .
$\text{HCF} = 2 \times 3 \times 7 = 42$

Try it

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

Finding the HCF of three numbers

Find the HCF of $72$ , $108$ and $180$ .

$72 = 2^3 \times 3^2$
$108 = 2^2 \times 3^3$
$180 = 2^2 \times 3^2 \times 5$
Common prime factors: $2$ and $3$ (the prime $5$ appears only in $180$ , so it is excluded).
Take the smallest index for each: $2^2$ and $3^2$ .
$\text{HCF} = 2^2 \times 3^2 = 36$

Practise this

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

Practise Primes, HCF and LCM