Every integer is a unique product of primes
Break a number down with a factor tree, then write it as a product of primes in index form, e.g. .
HCF: lowest powers; LCM: highest powers
From the prime factorisations: the HCF (highest common factor) multiplies the lowest power of each common prime; the LCM (lowest common multiple) multiplies the highest power of every prime that appears.
Drawn from real examiner reports.
Confusing HCF and LCM
Students mix up the two — taking the highest powers for the HCF (highest common factor) or the lowest for the LCM (lowest common multiple) — or cannot apply the idea when the numbers are given in index form.
Many students confused the two terms on Nov 2024 Paper 1H (Q6).
Wrong powers of the common primes
Picking common primes but using the wrong power, e.g. giving instead of for an HCF (highest common factor).
Less careful students made exactly this error on Nov 2024 Paper 1H (Q6).
Giving a value, not a product of primes
When the question says "as a product of prime factors", multiplying out to a single number (e.g. writing 700) loses the form mark; some also include a prime that is not common, e.g. an extra .
Writing 700 with no working scored only 1 mark on Nov 2024 Paper 1H (Q6); the summary advised students to "write a number as a product of its primes in index form".
Stopping the factor tree too early
A branch such as or is not prime, so the tree is unfinished: split and . Leaving a composite in the final product (e.g. ) means the number is not fully written as a product of primes and drops the accuracy mark.
(general exam technique)
Treating 1 as a prime factor
The number is not prime, so it never appears in a prime factorisation — writing adds nothing and signals a misunderstanding of what a prime is. Every prime factor is or more, and is the only even prime.
(general exam technique)
Answer in the form the question asks for
If it says "as a product of prime factors", leave it in index form — do not evaluate it to a single number.
Self-check with HCF times LCM
Verify that , the product of the two original numbers. For and : , so the pair is consistent before you commit.
Lay the factorisations out in a grid
List each prime once and write its power in each number. Reading down the columns, the HCF takes the lower power and the LCM the higher — a layout that stops you swapping the two rules under time pressure.
Use a factor tree (or repeated division) to break a number into primes, then collect them in index form: If a question asks for a number "as a product of its prime factors", leave it in this form — don't multiply it back out.
"As a product of prime factors" — what must your answer look like?
Write as a product of its prime factors, giving your answer in index form.
