Factors, multiples and primes: prime factor decomposition (product of prime factors), the highest common factor (HCF) and lowest common multiple (LCM), and using Venn diagrams to find them.

What this dot point is asking

Edexcel expects you to break a number into its prime factors, and to use that decomposition to find the highest common factor (HCF) and lowest common multiple (LCM) of two or more numbers. These ideas underpin simplifying fractions, working with ratios and solving "when do two events coincide" problems, so they recur far beyond this topic.

Factors, multiples and primes

The three words are easy to mix up, so fix them early.

Prime factor decomposition

Every integer greater than $1$ can be written as a product of primes in exactly one way. A factor tree is the standard method: split the number into any factor pair, then keep splitting until every branch ends in a prime.

For $84$: $84 = 4 \times 21 = (2 \times 2) \times (3 \times 7)$, so $84 = 2^2 \times 3 \times 7$. Always present the answer in index form and check that every factor is genuinely prime.

Finding the HCF and LCM from prime factors

Once both numbers are written as products of primes, the HCF and LCM follow from simple rules.

For $48 = 2^4 \times 3$ and $60 = 2^2 \times 3 \times 5$: the HCF takes the lowest shared powers, $2^2 \times 3 = 12$; the LCM takes the highest powers of all primes, $2^4 \times 3 \times 5 = 240$. The check works: $12 \times 240 = 2880 = 48 \times 60$.

Using a Venn diagram

A Venn diagram organises the prime factors visually and is the method Edexcel often expects.

LCM problems often appear as word problems, such as two buses leaving together and asking when they next coincide. The answer is the LCM of their intervals.

Why prime factors are so useful

The reason prime factorisation appears so often is that it exposes the "building blocks" of a number, and many other techniques rely on those blocks. Simplifying a fraction such as $\dfrac{48}{60}$ is just cancelling the HCF: both share $2^2 \times 3 = 12$, so the fraction reduces to $\dfrac{4}{5}$. Adding fractions needs the LCM of the denominators as the common denominator. Working out whether a number is a perfect square is instant from its prime factors: $900 = 2^2 \times 3^2 \times 5^2$ has every power even, so $\sqrt{900} = 2 \times 3 \times 5 = 30$. Even surd simplification uses the same idea, since $\sqrt{72} = \sqrt{2^3 \times 3^2} = 3 \times 2\sqrt{2} = 6\sqrt 2$.

Try this

Q1. Write $126$ as a product of its prime factors in index form. [2 marks]

• Cue. $126 = 2 \times 63 = 2 \times 9 \times 7 = 2 \times 3^2 \times 7$.

Q2. Two lighthouses flash every $12$ seconds and every $18$ seconds. They flash together now. After how many seconds do they next flash together? [2 marks]

• Cue. Find the LCM of $12$ and $18$: $12 = 2^2 \times 3$, $18 = 2 \times 3^2$, so the LCM is $2^2 \times 3^2 = 36$ seconds.