Cutting equal pieces uses the HCF. Finding when two schedules meet again uses the LCM. Learn four methods and how to choose the right one for any problem.
A tailor has two rolls of ribbon, 24 cm and 36 cm long, and wants to cut them into equal pieces with nothing wasted. Two traffic lights turn green together, then go out of step. When will they match again? Both questions have clean answers, and both use a pair of ideas that look alike but do opposite jobs: the highest common factor (HCF) and the lowest common multiple (LCM). This guide teaches four reliable methods and shows you which one to choose for a real problem.
A factor of a number divides it exactly, with nothing left over. The factors of 12 are 1, 2, 3, 4, 6 and 12. A multiple of a number is what you get by multiplying it by 1, 2, 3, 4 and so on. The multiples of 12 are 12, 24, 36, 48, 60 and so on, and they go on forever.
Here is the key difference: a number has only a few factors (all of them no bigger than the number), but it has endless multiples (all of them at least as big as the number). That is why the HCF is a “highest” and the LCM is a “lowest”. Factors have a top end, and multiples have a bottom end. To review these ideas, see Factors & Multiples and Divisibility Rules.
The highest common factor of two or more numbers is the largest number that divides all of them exactly. In some books it is called the GCF (greatest common factor) or GCD (greatest common divisor). They all mean the same thing.
Think of it as the biggest equal-sized piece that fits into each number a whole number of times, with nothing left over.
This is the easiest method for small numbers. List all the factors of each number, circle the ones they share, and pick the largest.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
Common factors: 1, 2, 3, 4, 6, 12.
Answer: HCF = 12
For larger numbers, listing every factor takes too long. Instead, break each number into its prime factors (see Factors & Multiples). Write repeated primes using powers. Then take each prime that appears in all the numbers, using the smallest power it has.
24 = 2 × 2 × 2 × 3 = 23 × 3. And 36 = 2 × 2 × 3 × 3 = 22 × 32.
Both contain 2 and 3. The smallest power of 2 is 22, and the smallest power of 3 is 31.
HCF = 22 × 3 = 4 × 3 = 12. This matches the listing method. ✓
Find the HCF of 84 and 120. 84 = 22 × 3 × 7. And 120 = 23 × 3 × 5.
Shared primes: 2 (smallest power 22) and 3 (power 1). The 7 and the 5 are not shared, so they are left out.
HCF = 22 × 3 = 12.
This method is quick even for huge numbers, because it needs no factors at all. Divide the bigger number by the smaller one. Then divide the smaller number by the remainder. Keep going until the remainder is 0. The last divisor is the HCF.
48 ÷ 18 = 2 remainder 12.
18 ÷ 12 = 1 remainder 6.
12 ÷ 6 = 2 remainder 0.
The last divisor, 6, is the HCF.
Find the HCF of 252 and 105.
252 ÷ 105 = 2 remainder 42. Then 105 ÷ 42 = 2 remainder 21. Then 42 ÷ 21 = 2 remainder 0.
HCF = 21. Check with primes: 252 = 22 × 32 × 7 and 105 = 3 × 5 × 7, so the shared part is 3 × 7 = 21. ✓
Picking the largest number in sight is a tempting shortcut, but it confuses the HCF with something else.
The fix: the HCF must divide every number in the problem exactly. Does 24 divide 36 exactly? No: 36 ÷ 24 = 1.5. The HCF can never be bigger than the smallest number in the problem. The correct HCF of 24 and 36 is 12, which divides both exactly (24 ÷ 12 = 2 and 36 ÷ 12 = 3).
Euclid’s method gets its name from the Greek mathematician Euclid, who described it in his book Elements more than 2,000 years ago. It is still used inside computers today.
The lowest common multiple of two or more numbers is the smallest number that all of them divide into exactly. It is the first time their multiples “meet”.
Multiples of 12: 12, 24, 36, 48, 60, 72, …
Multiples of 18: 18, 36, 54, 72, …
The first number in both lists is 36. LCM = 36. (The next shared multiple is 72, which is also a common multiple, but not the lowest.)
This time, take every prime that appears in any of the numbers, using the largest power it has.
24 = 23 × 3 and 36 = 22 × 32. The largest power of 2 is 23, and the largest power of 3 is 32.
LCM = 23 × 32 = 8 × 9 = 72.
84 = 22 × 3 × 7 and 120 = 23 × 3 × 5. Take all primes, with the largest power of each: 23, 3, 5 and 7.
LCM = 8 × 3 × 5 × 7 = 840.
Multiplying the two numbers always gives a common multiple, but not always the lowest one.
The fix: 24 is a common multiple of 4 and 6, but 12 is smaller and is also a multiple of both (12 ÷ 4 = 3 and 12 ÷ 6 = 2). The LCM is 12. With primes: 4 = 22 and 6 = 2 × 3, so LCM = 22 × 3 = 12. Multiplying the two numbers is only the LCM when they share no factor other than 1.
For two numbers, there is a neat link between their HCF and LCM:
HCF × LCM = first number × second number
For 24 and 36: HCF = 12 and LCM = 72, so 12 × 72 = 864, and 24 × 36 = 864. ✓
So if you know the HCF, you can find the LCM quickly: LCM = (24 × 36) ÷ 12 = 72. For 15 and 20, the HCF is 5, so LCM = (15 × 20) ÷ 5 = 60.
Important: this shortcut works only for two numbers. For three or more numbers it does not hold.
The prime factor method extends naturally to three or more numbers.
12 = 22 × 3. And 18 = 2 × 32. And 30 = 2 × 3 × 5.
HCF: only 2 and 3 appear in all three, with smallest powers 21 and 31. HCF = 2 × 3 = 6.
LCM: take every prime with its largest power: 22, 32 and 5. LCM = 4 × 9 × 5 = 180.
Check that the shortcut fails here: 12 × 18 × 30 = 6,480, but HCF × LCM = 6 × 180 = 1,080. They are not equal, so the product rule is for two numbers only.
Two numbers are co-prime (or relatively prime) when their only common factor is 1, so their HCF is 1. The numbers do not have to be prime themselves. For example, 8 and 15 are both composite, but they share no factor except 1.
HCF of 8 and 15 = 1, so they are co-prime. Their LCM is simply 8 × 15 = 120, since for co-prime numbers the LCM equals the product.
Tap any pair of numbers below to see their factors, common factors, HCF, and LCM all at once.
Tap a pair above to see the full breakdown.
Real problems do not say “find the HCF”. You must decide. Use these clues:
| Use the HCF when… | Use the LCM when… |
|---|---|
| You are splitting or cutting into equal groups or pieces | You are waiting for events to happen together again |
| You want the largest possible size or the fewest groups | You want the smallest number or the first time |
| Words like “largest”, “greatest”, “equal groups”, “no leftovers” | Words like “smallest”, “next time together”, “every … minutes” |
| The answer is no bigger than the smallest number given | The answer is no smaller than the biggest number given |
Cutting two ribbons, 24 cm and 36 cm long, into the largest possible equal pieces with none wasted uses the HCF: pieces of 12 cm each. Two traffic lights that turn green together, one every 12 seconds and the other every 18 seconds, will next turn green together after the LCM: every 36 seconds.
A shop has 48 pencils and 72 erasers. It makes identical gift packs with nothing left over. What is the greatest number of packs, and what does each pack hold?
“Greatest number of identical packs, none left” points to the HCF. HCF(48, 72): 48 = 24 × 3 and 72 = 23 × 32, so HCF = 23 × 3 = 24.
Answer: 24 packs. Each pack holds 48 ÷ 24 = 2 pencils and 72 ÷ 24 = 3 erasers.
A floor is 360 cm long and 240 cm wide. What is the side of the largest square tile that fits exactly, and how many tiles are needed?
HCF(360, 240) = 120 (360 = 23 × 32 × 5, 240 = 24 × 3 × 5, so HCF = 23 × 3 × 5 = 120).
Answer: 120 cm tiles. Tiles along the length: 360 ÷ 120 = 3. Along the width: 240 ÷ 120 = 2. Total: 3 × 2 = 6 tiles.
Two bus routes leave a station at 7:00 am. Bus A leaves every 15 minutes, and bus B every 20 minutes. When do they next leave together?
“Next time together” points to the LCM. LCM(15, 20) = 60 (15 = 3 × 5 and 20 = 22 × 5, so LCM = 22 × 3 × 5 = 60).
Answer: after 60 minutes, at 8:00 am.
Find the smallest number that leaves a remainder of 3 when divided by 4, by 6 or by 15.
First find the LCM of 4, 6 and 15: 4 = 22, 6 = 2 × 3, 15 = 3 × 5, so LCM = 22 × 3 × 5 = 60. A number leaving remainder 3 for all three must be 3 more than a multiple of 60. The smallest is 60 + 3 = 63.
Check: 63 ÷ 4 = 15 remainder 3, 63 ÷ 6 = 10 remainder 3, 63 ÷ 15 = 4 remainder 3. ✓
Three bells ring every 6, 8 and 12 minutes. They ring together at noon. When do they next ring together?
LCM(6, 8, 12) = 24 (6 = 2 × 3, 8 = 23 and 12 = 22 × 3, so LCM = 23 × 3 = 24). Answer: after 24 minutes, at 12:24 pm.
HCF and LCM are the hidden tools inside fraction arithmetic. Simplifying a fraction means dividing top and bottom by their HCF. Adding fractions with different denominators uses the LCM as the common denominator.
(a) Simplify 36/60. HCF(36, 60) = 12, so 36/60 = (36 ÷ 12)/(60 ÷ 12) = 3/5.
(b) Work out 5/12 + 7/18. LCM(12, 18) = 36. 5/12 = 15/36 and 7/18 = 14/36. Sum = 29/36.
This is explained in more detail in Fraction Fix-It.
The Euclidean algorithm for finding the HCF is one of the oldest algorithms still in everyday use. Its idea is also inside modern computers, where it helps with tasks such as simplifying fractions and certain kinds of data security.
Tap the correct answer for each question. Your score is tracked as you go, and your best score is saved on this device.
1. 18 = 2 × 32 and 24 = 23 × 3. HCF = 2 × 3 = 6. LCM = 23 × 32 = 72. (Check: 6 × 72 = 432 = 18 × 24. ✓)
2. 91 ÷ 35 = 2 remainder 21. 35 ÷ 21 = 1 remainder 14. 21 ÷ 14 = 1 remainder 7. 14 ÷ 7 = 2 remainder 0. HCF = 7.
3. LCM(10, 25): 10 = 2 × 5 and 25 = 52, so LCM = 2 × 52 = 50 seconds.
1. 60 = 22 × 3 × 5 and 84 = 22 × 3 × 7. HCF = 22 × 3 = 12. LCM = 22 × 3 × 5 × 7 = 420.
2. 252 = 2 × 105 + 42. Then 105 = 2 × 42 + 21. Then 42 = 2 × 21 + 0. HCF = 21.
3. HCF × LCM = product, so the other number = (6 × 180) ÷ 30 = 36.
4. Work in centimetres: 420 cm by 360 cm. HCF(420, 360): 420 = 22 × 3 × 5 × 7 and 360 = 23 × 32 × 5, so HCF = 22 × 3 × 5 = 60. The largest tile has a side of 60 cm. Tiles needed = (420 ÷ 60) × (360 ÷ 60) = 7 × 6 = 42 tiles.
Learners often swap HCF and LCM because both words sound alike. Give them physical or drawn tasks first: cut paper strips into equal pieces for the HCF, and mark multiples on two number lines for the LCM, as in the diagrams above. Then ask them to decide, before any calculation, whether the answer should be smaller or larger than the given numbers. This single check prevents most mix-ups. After the methods are secure, introduce word problems and ask students to say aloud why they chose HCF or LCM.
HCF is how fractions are simplified to lowest terms. LCM is how fractions with different denominators are added or subtracted, because finding a common denominator is finding an LCM. Beyond fractions, HCF and LCM appear whenever two repeating patterns or divisible groups must be compared: timetables, packaging, tiling and construction all rely on these two ideas.
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