Primes, HCF & LCM
Every number is built from primes. Once you can see those building blocks, HCF and LCM stop being two rules to memorise and become one picture.
What makes a number prime
A prime number has exactly two factors: 1 and itself. 7 is prime because only 1 and 7 divide it. 9 is not, because 3 divides it too.
Numbers with more than two factors are composite. So 9, 12 and 100 are all composite.
And 1 is neither. It has only one factor. That is not a technicality invented to annoy you. It is what makes the next idea work. If 1 counted as prime, then 12 could be written as 2 × 2 × 3, or 1 × 2 × 2 × 3, or 1 × 1 × 2 × 2 × 3, and "the" prime factorisation of a number would no longer be one single thing.
2 is the only even prime. Every other even number has 2 as a third factor, so it is out.
Finding the primes
You do not have to test every number. Cross out the multiples of 2, then of 3, then of 5, then of 7, whatever survives below 50 is prime. This is the sieve of Eratosthenes, and it is over two thousand years old.
Why stop testing at 7? Because 7 × 7 = 49, and any composite number below 50 must have a factor at or below its own square root. If nothing up to 7 divides it, nothing will.
Worth knowing by sight: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Those ten come up constantly.
Breaking a number into primes
Split the number into any two factors, then keep splitting until everything is prime. The order you split in does not matter. You always land on the same primes. That result has a name: the fundamental theorem of arithmetic, which says every whole number above 1 has exactly one prime factorisation.
Write repeats in index form: 60 = 2² × 3 × 5 rather than 2 × 2 × 3 × 5. It is shorter, and it is what makes HCF and LCM quick.
Try it two ways. Start 60 as 6 × 10, or start it as 4 × 15, or as 2 × 30. All three finish at 2² × 3 × 5.
HCF and LCM are one picture
Put the prime factors of both numbers into a Venn diagram.
- The HCF (highest common factor) is what sits in the overlap: the primes both numbers own.
- The LCM (lowest common multiple) is everything in the picture, overlap counted once.
That is why HCF × LCM = the product of the two numbers: multiply them together and you have counted the overlap twice, which is exactly once too many. Check it: 6 × 1260 = 7560, and 84 × 90 = 7560.
This gives you a free shortcut. If you know the HCF and one number, the other is HCF × LCM ÷ the number you have.
Which one does the question want?
Almost every word problem on this topic is one of two shapes, and the giveaway is whether things are being cut up or coming round again.
HCF: splitting into equal groups. "Ribbons 72 cm and 120 cm are cut into equal pieces with none left over. What is the longest a piece can be?" You need a length that divides both, and you want the biggest one. HCF(72, 120) = 24 cm. Same shape: tiling a floor with the largest square tile, or packing two kinds of item into identical boxes.
LCM: cycles meeting again. "Two bells ring together, then every 8 and every 12 minutes, when do they next coincide?" You need a time that both divide into, and you want the earliest. LCM(8, 12) = 24 minutes. Same shape: buses leaving a stop, or lights flashing together.
One sentence to hold onto: HCF is the biggest thing that fits inside both. LCM is the smallest thing both fit inside.
Where marks get lost
- Calling 1 prime. It has one factor, not two.
- Calling 2 composite because it is even. Even is not the test; the number of factors is.
- Stopping the factor tree early. 60 = 6 × 10 is not an answer: 6 and 10 are both still composite.
- Swapping HCF and LCM. The HCF is never bigger than the smaller number; the LCM is never smaller than the larger one. If your answer breaks that, you have them the wrong way round.
- Assuming the LCM is the two numbers multiplied. That is only true when the HCF is 1. LCM(4, 6) is 12, not 24.
- Guessing 91 is prime. It is 7 × 13. So is 51 (3 × 17) and 87 (3 × 29). Test with the small primes before you commit.
What to remember
- A prime has exactly two factors. 1 has one, so it is not prime. 2 is the only even prime.
- Every number above 1 has one prime factorisation. Write it in index form.
- HCF = shared primes. LCM = all the primes.
- HCF × LCM = the product of the two numbers.
- Cutting into equal parts → HCF. Cycles meeting again → LCM.
- Sanity check every answer: HCF ≤ smaller number ≤ larger number ≤ LCM.
Watch one worked through
Here is one done slowly, with the reason for every move. Then one for you.
Find the HCF and the LCM of 12 and 18.
- Break each number into prime factors.12 = 2 × 2 × 3 18 = 2 × 3 × 3Prime factors are the building blocks. Once both numbers are in this form, both answers can be read off.
- For the HCF take the primes that appear in both.common: 2 × 3 = 6Highest Common Factor means the biggest thing that divides both, so you keep only what they share.
- For the LCM take the most of each prime that appears in either.2 × 2 × 3 × 3 = 36Lowest Common Multiple must contain both numbers, so it needs enough of every prime for each of them.
- Check with the shortcut.HCF × LCM = 6 × 36 = 216 12 × 18 = 216 ✓HCF times LCM always equals the product of the two numbers. A free check on every question of this kind.
Answer: HCF = 6, LCM = 36
What is the HCF of 8 and 20?
Show the answer
4
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