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 lowest powers, LCM uses 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.
A commonly reported error: students confuse the two terms.
Wrong powers of the common primes
Picking the common primes correctly but using the wrong power, e.g. giving instead of for an HCF (highest common factor). Keep the lowest power of each shared prime, not just the bare prime.
Less careful students make exactly this error.
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 — leave it in index form such as .
Writing the evaluated number with no working is typically capped at 1 mark; examiners advise students to "write a number as a product of its primes in index form".
Putting a non-shared prime in the HCF
An HCF (highest common factor) uses only primes that appear in every number. Slipping in a prime found in just one of them — an extra , or a that only one number carries — makes the answer too big.
Not scaling 4A or 3B before combining
If a number is given as or , fold the multiplier into the prime factorisation first — e.g. raises the power of by — then read off the powers. Finding the HCF or LCM of and instead is a classic trap.
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 HCF × LCM = product of the numbers
For two numbers, equals their product. Use it to check an answer, or to find a missing number when the HCF, LCM and one of the numbers are given.
Spot LCM word-problems
Phrases like "ring/flash/leave together again" (bells, lights, buses) ask for the LCM — the next shared multiple. "Largest that divides both" asks for the HCF.
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?
One lighthouse flashes every seconds and another every seconds. They flash together at noon. After how many seconds do they next flash together?
