In this guide
LCM and HCF questions are short, and most of them can be done in under a minute. The hard part is not the calculation. It is reading a word problem and deciding which of the two it wants. Once you know the handful of question types, that decision becomes automatic.
This topic also feeds other chapters. You use LCM to add fractions and to set up the "total work" method in time and work, and HCF to simplify fractions and ratios. So a week spent here pays back later.
What the two words mean
HCF (Highest Common Factor) is the largest number that divides all the given numbers exactly. It is also called the GCD (greatest common divisor).
LCM (Lowest Common Multiple) is the smallest number that all the given numbers divide into exactly.
Take 12 and 18. The factors of 12 are 1, 2, 3, 4, 6, 12, and the factors of 18 are 1, 2, 3, 6, 9, 18. The largest common one is 6, so HCF = 6. The multiples of 12 are 12, 24, 36, 48… and of 18 are 18, 36, 54… The first common one is 36, so LCM = 36.
| Point | HCF | LCM |
|---|---|---|
| Size | Never more than the smallest number | Never less than the largest number |
| From prime factors | Common primes, lowest power | All primes, highest power |
| Word-problem clue | Largest, greatest, maximum size, equal groups | Together again, smallest number, least, minimum |
| Relationship | HCF always divides the LCM | LCM is always a multiple of the HCF |
Method 1: prime factorisation
Break each number into primes, then:
- HCF = product of the primes common to all, each taken the fewest times it appears.
- LCM = product of every prime that appears, each taken the most times it appears.
For 24 and 36: 24 = 2 × 2 × 2 × 3 = 2³ × 3 and 36 = 2 × 2 × 3 × 3 = 2² × 3².
- HCF = 2² × 3 = 12.
- LCM = 2³ × 3² = 8 × 9 = 72.
Method 2: division
HCF by repeated division. Divide the larger number by the smaller. Then divide the previous divisor by the remainder. Keep going until the remainder is 0. The last divisor is the HCF. This is quick for two large numbers.
LCM by the ladder. Write the numbers in a row and keep dividing by a prime that divides at least two of them. Carry down any number that is not divisible. For 12, 15 and 20:
| Divide by | Row of numbers |
|---|---|
| Start | 12, 15, 20 |
| 2 | 6, 15, 10 |
| 2 | 3, 15, 5 |
| 3 | 1, 5, 5 |
| 5 | 1, 1, 1 |
LCM = 2 × 2 × 3 × 5 = 60.
The product rule
For two numbers only: HCF × LCM = first number × second number.
This does not work for three numbers. Use it only when the question gives exactly two.
The six common word problems
| The question says | Use | What to do |
|---|---|---|
| Bells, lights or runners meet "together again" | LCM | LCM of the intervals |
| Largest tile, rope piece or container that fits exactly | HCF | HCF of the lengths |
| Smallest number divisible by all of them | LCM | LCM of the divisors |
| Smallest number leaving the same remainder r in each case | LCM | LCM + r |
| Greatest number that divides each, leaving remainder r | HCF | HCF of (each number − r) |
| Largest equal groups with nothing left over | HCF | HCF of the quantities |
Solved examples
Example 1. Find the HCF and LCM of 84 and 126.
- HCF by division: 126 ÷ 84 = 1, remainder 42.
- Next, 84 ÷ 42 = 2, remainder 0. So HCF = 42.
- LCM by the product rule: 84 × 126 ÷ 42 = 2 × 126 = 252.
- Check: 252 ÷ 84 = 3 and 252 ÷ 126 = 2.
Example 2. The HCF of two numbers is 6 and their LCM is 180. One number is 36. Find the other.
- HCF × LCM = product of the two numbers.
- 6 × 180 = 36 × other, so 1,080 = 36 × other.
- Other = 1,080 ÷ 36 = 30.
- Check: HCF(36, 30) = 6 and LCM(36, 30) = 180.
Example 3. Three bells ring every 6, 8 and 12 minutes. They ring together at 8:00 a.m. When will they next ring together, and how many times will they ring together from 8:00 a.m. to 10:00 a.m., both included?
- LCM(6, 8, 12) = 24 minutes, so the next time is 8:24 a.m.
- Two hours = 120 minutes, and 120 ÷ 24 = 5 gaps.
- Counting 8:00 itself: 5 + 1 = 6 times.
Example 4. Find the smallest number that leaves remainder 5 when divided by 12, 15 or 20.
- LCM(12, 15, 20) = 60 (from the ladder above).
- Add the remainder: 60 + 5 = 65.
- Check: 65 = 12 × 5 + 5 = 15 × 4 + 5 = 20 × 3 + 5.
Example 5. Find the greatest number that divides 70 and 94, leaving remainder 4 in each case.
- Remove the remainder: 70 − 4 = 66 and 94 − 4 = 90.
- HCF(66, 90): 66 = 2 × 3 × 11 and 90 = 2 × 3² × 5, so HCF = 2 × 3 = 6.
- Check: 70 = 6 × 11 + 4 and 94 = 6 × 15 + 4.
Example 6. A floor measures 6 m 24 cm by 4 m 32 cm. It is to be covered with identical square tiles, none cut. Find the largest tile size and the number of tiles.
- Convert to one unit: 624 cm and 432 cm.
- 624 = 2⁴ × 3 × 13 and 432 = 2⁴ × 3³, so HCF = 2⁴ × 3 = 48 cm.
- Tiles = (624 ÷ 48) × (432 ÷ 48) = 13 × 9 = 117 tiles.
Common mistakes
| Mistake | Correct idea |
|---|---|
| Using LCM for "largest tile" | Largest size that fits exactly is always HCF |
| Using the product rule for three numbers | It holds for two numbers only |
| Mixing metres and centimetres | Convert everything to one unit first |
| Forgetting the start time when counting "how many times" | Add 1 for the first ring |
| Subtracting the remainder in LCM questions | For "leaves remainder r", add r to the LCM |
Practice set
- Find the HCF of 36, 54 and 90.
- Find the LCM of 8, 12 and 18.
- The HCF of two numbers is 8 and their LCM is 96. One number is 24. Find the other.
- Two lights blink every 18 and 24 seconds. They blink together now. After how many seconds will they blink together again?
- Find the largest number that divides 60, 84 and 108 exactly.
- Find the smallest number that leaves remainder 2 when divided by 6, 9 or 15.
- Find the smallest four-digit number divisible by 12, 15 and 20.
- Two numbers are in the ratio 3 : 4 and their HCF is 5. Find their LCM.
Answers:
- 36 = 2² × 3², 54 = 2 × 3³, 90 = 2 × 3² × 5. Common: 2 × 3² = 18.
- 8 = 2³, 12 = 2² × 3, 18 = 2 × 3². LCM = 2³ × 3² = 72.
- 8 × 96 ÷ 24 = 32. Check: HCF(24, 32) = 8, LCM = 96.
- LCM(18, 24) = 72 seconds.
- HCF(60, 84, 108) = 12.
- LCM(6, 9, 15) = 90; 90 + 2 = 92.
- LCM = 60. 1,000 ÷ 60 = 16, remainder 40. Add 60 − 40 = 20, giving 1,020.
- The numbers are 3 × 5 = 15 and 4 × 5 = 20. LCM(15, 20) = 60.
What to do next
- Write the six-row word-problem table from memory, then check it.
- Find the HCF and LCM of 10 pairs of numbers by both methods.
- Revise factors and primes in our number system guide if factorising feels slow.
- Use LCM to add fractions in fractions and decimals, and to set up the units method in time and work.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Staff Selection Commission website .
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