HCF and LCM

El método, paso a paso

1. 1

   Write both numbers as products of primes

   36 = 2² × 3² · 48 = 2⁴ × 3

   Everything in this topic follows from these two lines. Students who try to find an HCF by listing factors succeed on small numbers and stall completely on numbers like 252 and 198, where the lists are long and easy to leave incomplete.

2. 2

   For the HCF, take the shared primes at the lower power

   shared: 2 (lower power 2²) and 3 (lower power 3¹) → 2² × 3 = 12

   A common factor must divide both, so it cannot use more of a prime than the poorer number has. Taking the lower power of each shared prime is exactly that constraint, expressed mechanically.

3. 3

   For the LCM, take every prime at the higher power

   2⁴ from 48, 3² from 36 → 2⁴ × 3² = 144

   A common multiple must contain both numbers, so it needs at least as much of every prime as the richer number has. Primes appearing in only one number are still included at their full power.

4. 4

   Or use a Venn diagram of the prime factors

   36 ∩ 48 = {2, 2, 3} → HCF = 12; all of 36 ∪ 48 → LCM = 144

   The overlap multiplies to the HCF and the whole diagram multiplies to the LCM. For students who find the index rules abstract, this makes both answers visible at once and comes from a single piece of work.

5. 5

   Check with the product rule

   HCF × LCM = 12 × 144 = 1728; 36 × 48 = 1728 ✓

   For any two numbers, the HCF multiplied by the LCM equals the product of the numbers. This is a complete check on both answers at once, and it takes one multiplication — easily the best value check in the topic.

Which one does the question want?

Most marks lost here are lost to finding the right number for the wrong quantity. The distinction is one of size and direction: the HCF is smaller than both numbers and is about splitting things up; the LCM is larger than both and is about things coming round again.

So a question about cutting ribbon into equal pieces, or arranging people into equal groups, wants the HCF. A question about two buses leaving together, or two lights flashing in step, wants the LCM. If your answer is bigger than both starting numbers and the question was about dividing something, you have found the wrong one.

• HCF — largest number dividing both. Smaller than both. Splitting up
• LCM — smallest number both divide into. Larger than both. Coming round again
• HCF: shared primes, lower power
• LCM: all primes, higher power
• Check: HCF × LCM = the product of the two numbers

Why listing does not scale

Listing factors works for 12 and 18 and collapses for 252 and 198. The lists are long, missing one is easy, and a single omission produces a confidently wrong answer with working that looks complete. Prime factorisation has no such failure mode: the method itself tells you when it has finished.

This is worth being firm about early, because students who found listing sufficient in Year 6 will resist the more abstract method until the numbers defeat them — usually in a GCSE exam rather than in a lesson.

Where these actually get used

The LCM is the lowest common denominator, which is the entire reason adding fractions works. Anyone who has found a common denominator has computed an LCM, usually without being told that is what it was called.

The HCF is what simplifies a fraction to its lowest terms in one step rather than several, and later what factorises an algebraic expression. Both keep appearing under other names, which is why the topic repays being learned properly rather than passed.

How we teach HCF and LCM

We teach the product check from the first lesson. Students who verify HCF × LCM against the product of the numbers catch swapped answers immediately, and swapping is by far the most common error in the topic.

We also anchor each one to a physical situation before any arithmetic — ribbon being cut for the HCF, buses departing for the LCM. The word problems in exams are almost always one of those two shapes, and a student who recognises the shape has done the hard part of the question.