In this guide
LCM and HCF questions are some of the most predictable in the RRB Group D maths section. There are only a few patterns: bells or runners meeting again, the largest tile or rope, a number that leaves a given remainder, and the product rule for two numbers. Learn to recognise the pattern from the wording, and each question becomes a two-line calculation.
What the words mean
- HCF (Highest Common Factor): the biggest number that divides all the given numbers exactly. It is also called GCD, the greatest common divisor.
- LCM (Lowest Common Multiple): the smallest number that all the given numbers divide exactly.
For 12 and 18: the common factors are 1, 2, 3 and 6, so HCF = 6. The common multiples are 36, 72, 108 and so on, so LCM = 36.
A quick sense check: the HCF is never bigger than the smallest number, and the LCM is never smaller than the biggest number.
Method 1: prime factors
- Break each number into primes.
- HCF = the primes common to all, each with its lowest power.
- LCM = every prime that appears, each with its highest power.
Our number system guide covers prime factors if you need a refresher.
Method 2: division, for the HCF of large numbers
When numbers are large and hard to factorise, divide the larger by the smaller, then divide the last divisor by the remainder, and keep going until the remainder is 0. The last divisor is the HCF.
For the HCF of 391 and 667:
- 667 = 391 × 1 + 276
- 391 = 276 × 1 + 115
- 276 = 115 × 2 + 46
- 115 = 46 × 2 + 23
- 46 = 23 × 2 + 0
The HCF is 23. Check: 391 = 23 × 17 and 667 = 23 × 29.
The product rule
For two numbers only: HCF × LCM = the product of the two numbers.
This is behind a very common question: "The HCF of two numbers is 6, the LCM is 72, and one number is 18. Find the other." The answer is 6 × 72 ÷ 18 = 24.
Which one does the question want?
| Words in the question | Use | Why |
|---|---|---|
| "together again", "at the same time", "ring together" | LCM | You need a time that fits every cycle |
| "smallest number divisible by", "least number" | LCM | You need a common multiple |
| "greatest", "longest", "largest size", "maximum length" | HCF | You need the biggest common measure |
| "leaves the same remainder" in a smallest-number question | LCM + remainder | Build the multiple, then add |
| "leaves remainders" in a greatest-number question | HCF of the differences | Remove the remainders first |
Worked examples
Example 1: Find the HCF and LCM of 24 and 60.
- 24 = 2³ × 3; 60 = 2² × 3 × 5.
- HCF = 2² × 3 = 12.
- LCM = 2³ × 3 × 5 = 120.
- Check with the product rule: 12 × 120 = 1,440 = 24 × 60.
Example 2: Three runners complete a lap in 12, 15 and 20 minutes. They start together. When will they next be at the start together?
- 12 = 2² × 3; 15 = 3 × 5; 20 = 2² × 5.
- LCM = 2² × 3 × 5 = 60 minutes.
Example 3: What is the longest rope that can measure 84 m, 126 m and 210 m exactly?
- 84 = 2² × 3 × 7; 126 = 2 × 3² × 7; 210 = 2 × 3 × 5 × 7.
- HCF = 2 × 3 × 7 = 42 m.
- Check: 84 ÷ 42 = 2, 126 ÷ 42 = 3, 210 ÷ 42 = 5.
Example 4: Find the smallest number which, when divided by 8, 12 and 18, leaves remainder 5 each time.
- 8 = 2³; 12 = 2² × 3; 18 = 2 × 3².
- LCM = 2³ × 3² = 72.
- Add the remainder: 72 + 5 = 77.
- Check: 77 ÷ 8 = 9 remainder 5; 77 ÷ 12 = 6 remainder 5; 77 ÷ 18 = 4 remainder 5.
Example 5: Find the greatest number that divides 70 and 125, leaving remainders 5 and 8 respectively.
- Take away the remainders: 70 − 5 = 65 and 125 − 8 = 117. The number must divide both exactly.
- 65 = 5 × 13; 117 = 3² × 13.
- HCF = 13.
- Check: 70 = 13 × 5 + 5 and 125 = 13 × 9 + 8.
Example 6 (fractions): Find the HCF and LCM of 2/3 and 4/9.
- HCF of fractions = HCF of the numerators ÷ LCM of the denominators = HCF(2, 4) ÷ LCM(3, 9) = 2 ÷ 9 = 2/9.
- LCM of fractions = LCM of the numerators ÷ HCF of the denominators = LCM(2, 4) ÷ HCF(3, 9) = 4 ÷ 3 = 4/3.
Common mistakes
- Using LCM for a "greatest" or "longest" question, or HCF for a "together again" question.
- Using the product rule for three numbers.
- Forgetting to add the remainder in a smallest-number question.
- In a greatest-number question, forgetting to subtract the remainders before finding the HCF.
- Mixing units, such as metres and centimetres, before finding the HCF.
Practice set
- Find the LCM of 6, 10 and 15.
- Find the HCF of 36 and 90.
- The HCF of two numbers is 6 and their LCM is 72. One number is 18. Find the other.
- Three bells ring every 4, 6 and 8 minutes. They ring together now. After how long will they ring together again?
- A floor is 360 cm by 480 cm. What is the largest square tile that fits it exactly, and how many tiles are needed?
- Find the smallest four-digit number divisible by 12, 15 and 20.
- Find the greatest number that divides 62 and 98, leaving remainder 2 in each case.
- Two numbers are in the ratio 3 : 4 and their HCF is 5. Find their LCM.
Answers:
- 30. 6 = 2 × 3, 10 = 2 × 5, 15 = 3 × 5; LCM = 2 × 3 × 5.
- 18. 36 = 2² × 3², 90 = 2 × 3² × 5; HCF = 2 × 3².
- 24. 6 × 72 ÷ 18 = 432 ÷ 18 = 24.
- 24 minutes. LCM of 4, 6, 8 = 2³ × 3 = 24.
- 120 cm tiles, 12 of them. HCF of 360 and 480 is 120; 360 ÷ 120 = 3 and 480 ÷ 120 = 4, so 3 × 4 = 12 tiles.
- 1,020. LCM of 12, 15, 20 = 60. 1,000 ÷ 60 = 16 remainder 40, so the next multiple is 60 × 17 = 1,020.
- 12. Subtract the remainder: 60 and 96. HCF(60, 96) = 12. Check: 62 = 12 × 5 + 2, 98 = 12 × 8 + 2.
- 60. The numbers are 3 × 5 = 15 and 4 × 5 = 20. LCM(15, 20) = 60.
What to do next
- Copy the "which one does the question want?" table into your notebook.
- Practise the division method on two pairs of three-digit numbers.
- Solve 15 LCM–HCF questions from past papers and label each one by pattern.
- Check where LCM–HCF sits in the maths plan, then move to the next topic.
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