Factors, HCF and LCM

Why this matters

Factors sit underneath almost every calculation with fractions. Simplifying a fraction, finding a common denominator and cancelling before you multiply all use them — and so do practical problems like cutting lengths of material with no waste.

Two ways to find the HCF

Listing. Write out every factor of both numbers and find the biggest one on both lists. It works, but it is easy to miss factors (Harry listed the factors of 60 as 1, 2, 5, 6, 12, 30, 60 and forgot 3, 4, 10, 15 and 20).

Prime factors. Build a factor tree for each number and multiply the shared primes. This is faster and far more reliable for big numbers.

Number words you need

Word	Meaning	Examples
Prime	Exactly two factors	2, 3, 5, 7, 11, 13, 17, 19
Composite	More than two factors	4, 6, 8, 9, 10
Square	Has an odd number of factors	1, 4, 9, 16

1 is neither prime nor composite — its only factor is 1.

Simplifying fractions

Divide the numerator and the denominator by their HCF and the fraction is in its simplest form in one step:

$\frac{48}{112} = \frac{48 \div 16}{112 \div 16} = \frac{3}{7}$

Handy check. A fraction is fully simplified when the only common factor of the top and bottom is 1.

Key rules and formulas

Factors come in pairs

Divide by 1, 2, 3, … in turn. Each time it divides exactly you get two factors:

36 = 1 \times 36 = 2 \times 18 = 3 \times 12 = 4 \times 9 = 6 \times 6

. Stop when the pairs meet.

Prime factors

$30 = 2 \times 3 \times 512 = 2 \times 2 \times 3 = 2^{2} \times 3$

A prime has exactly two factors, 1 and itself. Every whole number greater than 1 can be written as a product of primes in exactly one way.

HCF from prime factors

Multiply the primes the two numbers share.

12 = 2 \times 2 \times 3

and

30 = 2 \times 3 \times 5

2 \times 3

, so the HCF is 6.

LCM from prime factors

Take every prime the highest number of times it appears in either number. For 12 and 30:

2 \times 2 \times 3 \times 5 = 60

Worked examples

Worked example 1

Finding the HCF with prime factors

Find the HCF of 60 and 84.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Cancelling before multiplying

Work out

\frac{24}{49} \times \frac{35}{36}

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

A real problem

A garden measures 84 m by 112 m. It is split into the fewest possible identical square plots. How long is each side?

Try it yourself first, then reveal the solution one step at a time.

Numbers	HCF
20 and 30	Row 1, HCF
36 and 100	Row 2, HCF
36, 60 and 42	Row 3, HCF
126 and 180	Row 4, HCF

Numbers

HCF

20 and 30

36 and 100

36, 60 and 42

126 and 180

In this lesson you will learn to…

Why this matters

Two ways to find the HCF

Number words you need

Simplifying fractions

Key rules and formulas

Factors come in pairs

Prime factors

HCF from prime factors

LCM from prime factors

Worked examples

Finding the HCF with prime factors

Cancelling before multiplying

A real problem

Practice: factors, HCF and LCM