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Level 9Grade 9skill: hcf_lcm· 9 min read

HCF and LCM Explained

In short: Factorise each number into primes once and you get both answers: HCF is the shared primes at their lowest powers, LCM is every prime at its highest power. The exam trap is choosing between them - if the answer must divide the numbers it is HCF (equal groups, largest tile), and if it must be a multiple it is LCM (buses meeting, cycles lining up). For two numbers, HCF x LCM equals their product.

HCF and LCM: the method that always works

Every HCF or LCM question is the same question asked two ways. Break each number into its prime building blocks, and both answers fall out of the same list of primes — you just read that list differently.

The HCF (highest common factor) is the biggest number that divides all of them. Take the primes they share, each at its lowest power, and multiply. The LCM (lowest common multiple) is the smallest number they all divide into. Take every prime that appears, each at its highest power, and multiply.

The hard part of a Grade 9 paper is never the arithmetic — it is choosing which one you need. So before you compute anything, ask one question: does the answer have to divide the given numbers, or be a multiple of them? Equal shares, largest tile, longest piece → HCF. Cycles lining up, buses meeting, next time both ring → LCM.

  1. Write each number as a product of primes, e.g. 84 = 2 x 2 x 3 x 7.
  2. For HCF: keep only the primes common to every number, each at its lowest power, then multiply.
  3. For LCM: keep every prime that appears anywhere, each at its highest power, then multiply.
  4. Decide by meaning: must the answer divide the numbers (HCF) or be a multiple of them (LCM)?
  5. Sanity-check two numbers with HCF x LCM = the product of the two numbers.

Why one method gives you both answers Why it works

Splitting down -> HCF
If the answer must divide the given numbers exactly (identical groups, biggest tile, longest equal piece), you need the HCF. It can never be larger than the smallest number.

84 pencils and 126 erasers into identical kits with none left over -> HCF(84,126) = 42 kits.

Building up -> LCM
If the answer must be a multiple of the given numbers (repeating cycles coinciding, buses or bells meeting, the next common moment), you need the LCM. It can never be smaller than the largest number.

Buses leaving every 18 and 24 minutes next leave together after LCM(18,24) = 72 minutes.

The product law
For any two numbers, HCF x LCM = the product of the two numbers. Know any three of those four values and you can find the fourth.

HCF = 12, LCM = 240, one number = 48 -> other = (12 x 240) / 48 = 60.

HCF works downward: it splits your numbers into the biggest equal shares, so it is never bigger than the smallest number you started with. LCM works upward: it stacks cycles until they line up, so it is never smaller than the largest number. So if your answer came out bigger than the numbers you were given, you needed LCM; if it came out smaller, you needed HCF. That single sanity check catches almost every HCF-for-LCM mix-up before it costs you a mark.

Worked examples

Two buses leave the depot together at 9:00. Bus A returns and leaves every 18 minutes; Bus B every 24 minutes. At what time do they next leave together? = They next leave together at 10:12 (after 72 minutes).
  1. "Leave together again" means both cycles line up -> this is an LCM, not an HCF.
  2. Prime factorise: 18 = 2 x 32 and 24 = 23 x 3.
  3. LCM takes every prime at its highest power: 23 x 32 = 8 x 9 = 72.
  4. 72 minutes after 9:00 is 10:12.
  5. Trap check: HCF(18,24) = 2 x 3 = 6 would say "every 6 minutes" - far too soon, and smaller than both numbers, so it cannot be the meeting interval.
You have 84 pencils and 126 erasers. You want to build identical kits, each with the same number of pencils and the same number of erasers, using everything with nothing left over. What is the greatest number of kits, and what is in each? = 42 kits, each with 2 pencils and 3 erasers.
  1. "Identical kits, nothing left over" means the kit count must divide both totals -> this is an HCF.
  2. Prime factorise: 84 = 22 x 3 x 7 and 126 = 2 x 32 x 7.
  3. HCF takes shared primes at their lowest power: 2 x 3 x 7 = 42.
  4. Each kit gets 84 / 42 = 2 pencils and 126 / 42 = 3 erasers.
  5. Trap check: the LCM 22 x 32 x 7 = 252 is larger than either total - you cannot make 252 kits from 84 pencils.
Two numbers have an HCF of 12 and an LCM of 240. One of the numbers is 48. Find the other number. = The other number is 60.
  1. Use the product law: for two numbers, HCF x LCM = product of the numbers.
  2. So (the two numbers multiplied) = 12 x 240 = 2880.
  3. The other number = 2880 / 48 = 60.
  4. Verify: 48 = 24 x 3 and 60 = 22 x 3 x 5. HCF = 22 x 3 = 12 and LCM = 24 x 3 x 5 = 240. Both match.
Three ropes measure 108 cm, 144 cm and 180 cm. Each is cut into equal pieces, all the same length, with no rope wasted. If the pieces are as long as possible, how many pieces are there in total? = The pieces are 36 cm long, giving 12 pieces in total.
  1. "Equal pieces, no waste" means the piece length must divide all three lengths, and "as long as possible" means the greatest such length -> HCF.
  2. Prime factorise: 108 = 22 x 33, 144 = 24 x 32, 180 = 22 x 32 x 5.
  3. HCF takes shared primes at their lowest power: 22 x 32 = 36 cm per piece.
  4. Now answer the question actually asked - the number of pieces: 108/36 = 3, 144/36 = 4, 180/36 = 5.
  5. Total pieces = 3 + 4 + 5 = 12.
  6. Double trap: 36 is the piece length, not the count; and picking LCM here gives a length longer than the ropes.

Traps that turn a right method into a wrong answer

Using HCF when the problem needs LCM (or vice versa)

✗ For bells that ring every 15 and 20 minutes, computing HCF(15,20) = 5 and answering "they ring together every 5 minutes."

"Ringing together again" is two cycles lining up, so the answer must be a multiple of both 15 and 20 - that is the LCM. The HCF divides the numbers; it can never be the interval at which they coincide.

Fix: Ask whether the answer must divide the numbers (HCF) or be a multiple of them (LCM). Bells/buses meeting = LCM; equal groups/largest tile = HCF. Correct answer: LCM(15,20) = 22 x 3 x 5 = 60 minutes.

wrong_operation

Stopping at a common factor instead of the highest

✗ For HCF(48,72), spotting that both are divisible by 12 and answering 12.

12 is a common factor, but not the greatest one. Both numbers are still divisible by a larger value, so 12 is only a partial result. 48 = 24 x 3 and 72 = 23 x 32, so the HCF is 23 x 3 = 24.

Fix: After finding a common factor, keep going: use full prime factorisation and take every shared prime at its lowest power. Correct answer: HCF = 24.

partial_computation

Combining the prime factors with the wrong powers

✗ For LCM(18,24), writing 18 = 2 x 32 and 24 = 23 x 3, then multiplying 23 x 3 = 24 - grabbing the power of 3 from 24 instead of the higher power from 18.

The LCM needs the HIGHEST power of every prime. The higher power of 3 is 32 (from 18), not 31 (from 24). Taking the wrong power drops a factor of 3 and gives 24 instead of 72.

Fix: For each prime, scan all the numbers and take its largest exponent: 23 (from 24) and 32 (from 18). Correct answer: LCM = 23 x 32 = 72.

order_of_ops_lr

Dividing in the wrong order when checking a factor

✗ Checking whether 42 is the HCF of 84 and 126 by computing 42 / 84 = 0.5, deciding "that is not a whole number so 42 does not work," and dropping down to 21.

To test whether a number is a factor, you divide the big number BY the candidate, not the candidate by the big number. 84 / 42 = 2 and 126 / 42 = 3 are both whole, so 42 is a common factor - in fact the highest. Reversing the division hides that.

Fix: Always divide the given number by the candidate factor (number / factor). 21 does divide both, but it is not the greatest. Correct answer: HCF = 42.

reversed_operands

Practice HCF and LCM

Mix of equal-groups and meeting-cycle word problems, plus the HCF x LCM product law. Some answers divide the numbers, some are multiples - decide before you compute.

Keep going

First, make sure of

Frequently asked questions

What is the difference between HCF and LCM?

The HCF (highest common factor) is the biggest number that divides all the given numbers, so it is never larger than the smallest of them. The LCM (lowest common multiple) is the smallest number they all divide into, so it is never smaller than the largest of them. Both come from the same prime factorisation: HCF uses shared primes at their lowest powers, LCM uses every prime at its highest power.

How do I know whether a word problem needs HCF or LCM?

Ask whether the answer must divide the given numbers or be a multiple of them. Splitting things into equal groups, the largest tile, or the longest equal piece means the answer divides the numbers, so you need HCF. Repeating cycles coinciding - buses meeting, bells ringing together, the next common time - means the answer is a multiple, so you need LCM. Quick check: if your result is bigger than the numbers you started with, it should be an LCM; if smaller, an HCF.

Does HCF times LCM equal the product of the numbers for three numbers too?

No. HCF x LCM = the product of the numbers works only for exactly two numbers. For three or more it can fail, so do not rely on it there - factorise into primes and read off the HCF and LCM directly instead.

What happens if two numbers share no common factors?

Then their only common factor is 1, so the HCF is 1 and the numbers are called co-prime. In that case the LCM is simply the two numbers multiplied together. For example, HCF(8, 15) = 1 and LCM(8, 15) = 120.

What is the fastest way to find the HCF and LCM?

Prime factorisation handles both at once: list the primes, then take lowest shared powers for the HCF and highest powers for the LCM. For just the HCF of two large numbers, the Euclidean algorithm (repeatedly replace the larger number by its remainder when divided by the smaller) is quicker, and you can then get the LCM from HCF x LCM = the product.