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 uses the lowest powers; LCM uses the highest
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.
Choosing the 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).
Giving the value, not the product of primes (or wrong form)
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 .
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.
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.
