HCF and LCM Calculator With Prime Factors

Putting the numbers in

Choose how many numbers you have, from two to six, and type them in. They must be whole numbers of 1 or more, because prime factorisation is defined for whole numbers only; a decimal or a negative is reported and left out rather than rounded into something the question did not ask about.

Everything below the inputs comes from a single factorisation of each number. The HCF, the LCM, the Venn regions and the ladder are four presentations of the same arithmetic, not four separate calculations, which is why they cannot disagree with each other.

The upper limit of a million keeps trial division instant. It is far above anything a GCSE or IGCSE question will use — those numbers are chosen to factorise neatly by hand in under a minute.

Three routes to the same two answers

The factorisation route is the one mark schemes reward. Write each number in index form, then take the lowest power of each prime that appears in every number for the HCF, and the highest power of every prime that appears anywhere for the LCM. Getting those two rules the wrong way round is the single most common error on the topic.

The Venn route makes the rules visible rather than memorised. Each prime factor is placed in exactly one region: the overlap holds the copies every number shares, and each circle keeps the rest. Multiply the overlap and you have the HCF; multiply the whole diagram, each region once, and you have the LCM. This is also why the HCF can never be larger than the smallest number and the LCM can never be smaller than the largest.

The ladder route is repeated division. Divide the whole row by a prime they all share, write the results underneath, and repeat; the primes that divided all of them multiply to the HCF. To finish the LCM, carry on with any prime that divides at least two of them, then multiply every divisor down the side by whatever is left along the bottom.

• HCF — lowest power of the primes common to every number, the overlap of the Venn.
• LCM — highest power of every prime that appears in any of them, the whole Venn.
• Ladder — the same two answers by division, for anyone taught that method instead.

What this deliberately does not do

It does not draw a Venn diagram for four numbers or more. Two circles make three regions and three circles make seven, which is why school questions stop there; four sets need fifteen regions and a shape that is no longer two overlapping circles. Past three numbers the tool says so and falls back to showing what they all share and what each one keeps.

It does not print the HCF × LCM = a × b check for three numbers or more, because the identity is false there. The HCF of 4, 6 and 10 is 2 and the LCM is 60, which multiply to 120, while the three numbers multiply to 240. A tool that printed that check anyway would be teaching a rule that fails the first time a student relies on it.

And where the LCM is larger than a browser can hold as an exact whole number — possible with six large numbers, never with a textbook question — the exact answer is given in index form and the decimal is left off. A rounded LCM is not an LCM.

Where students go wrong

The confusion is almost never in the arithmetic. It is in deciding which of the two answers the question wants, and in remembering which rule goes with which.

A useful test on the wording: if the question is about things repeating until they line up again — two buses leaving together, bells ringing, runners meeting at the start line — the answer is the LCM, and it will be at least as large as the largest number given. If it is about breaking something into equal groups with nothing left over — cutting ribbon, packing boxes, arranging rows — it is the HCF, and it can be no larger than the smallest number given.

• Taking the highest power for the HCF and the lowest for the LCM, which reverses both answers.
• Including a prime in the HCF when it appears in only some of the numbers.
• Listing multiples until a common one appears, which works for 4 and 6 and collapses for 18 and 24.
• Leaving the answer as 2 × 2 × 2 × 3 where the question asked for index form.
• Assuming HCF × LCM = a × b works for three numbers, when it is a two-number identity.