1. Chapter at a Glance
Two buses leave a depot together — one every 12 minutes, the other every 18 minutes. When will they leave together again? A teacher wants to make equal groups from 24 boys and 36 girls with no one left out. What is the largest possible group size? Both questions are about what numbers have in common: common multiples and common factors. In this chapter you will find the HCF (highest common factor) and LCM (lowest common multiple) by several methods and use them to solve real problems.
- Common factors and the HCF.
- Common multiples and the LCM.
- Finding HCF and LCM by listing, by prime factorisation and by division.
- The relation HCF × LCM = product of two numbers.
- Choosing between HCF and LCM in word problems.
2. Common Factors and the HCF
A common factor of two numbers divides both of them. The greatest one is the HCF (also called GCD — greatest common divisor).
| Number | Factors |
|---|---|
| 24 | 1, 2, 3, 4, 6, 8, 12, 24 |
| 36 | 1, 2, 3, 4, 6, 9, 12, 18, 36 |
| Common | 1, 2, 3, 4, 6, 12 → HCF = 12 |
So 24 boys and 36 girls can be split into groups of at most 12 — 2 groups of boys and 3 groups of girls, everyone included.
- Every common factor is a factor of the HCF.
- If the HCF of two numbers is 1, they are co-prime.
- The HCF of two numbers is never larger than the smaller number.
3. Common Multiples and the LCM
A common multiple is divisible by both numbers. The smallest one is the LCM.
| Number | Multiples |
|---|---|
| 12 | 12, 24, 36, 48, 60, 72, … |
| 18 | 18, 36, 54, 72, 90, … |
| Common | 36, 72, 108, … → LCM = 36 |
So the two buses leave together again after 36 minutes.
- Every common multiple is a multiple of the LCM.
- The LCM is never smaller than the larger number.
- For co-prime numbers, LCM = product (LCM(8, 9) = 72).
4. HCF and LCM by Prime Factorisation
- Write each number as a product of primes (use powers).
- HCF: multiply the primes common to both, each taken with the smaller power.
- LCM: multiply all primes that appear, each taken with the larger power.
Find the HCF and LCM of 72 and 120.
Solution: 72 = 2³ × 3²; 120 = 2³ × 3 × 5. HCF = 2³ × 3 = 24. LCM = 2³ × 3² × 5 = 360. Check: 24 × 360 = 8640 = 72 × 120 ✔.
5. Division Methods
Common division for LCM
Write the numbers in a row and divide by primes that divide at least one of them, bringing down numbers that are not divisible. Stop when all become 1. The LCM is the product of the divisors.
Repeated subtraction / division for HCF (Euclid's idea)
The HCF of two numbers also divides their difference. So HCF(84, 36) = HCF(36, 84 − 36) = HCF(36, 48) = HCF(36, 12) = 12. Using division: 84 = 2 × 36 + 12, 36 = 3 × 12 + 0, so HCF = 12. This old method is very fast for large numbers.
6. The HCF–LCM Relation
This works only for two numbers, not three.
The HCF of two numbers is 6 and their LCM is 180. One number is 36. Find the other.
Solution: Other = 6 × 180 ÷ 36 = 30.
7. HCF or LCM? Choosing the Right Tool
| Key words | Use | Example |
|---|---|---|
| Largest size, greatest number, cut into equal pieces, arrange in rows exactly | HCF | Largest tile to cover a floor exactly |
| Smallest number, next time together, least length, ring together again | LCM | Bells ringing together |
8. Solved Problems (NCERT Style)
Three ribbons are 48 cm, 72 cm and 108 cm long. They must be cut into equal pieces of the greatest possible length with nothing wasted. Find that length and the number of pieces.
Solution: 48 = 2⁴ × 3, 72 = 2³ × 3², 108 = 2² × 3³. HCF = 2² × 3 = 12 cm. Pieces = 4 + 6 + 9 = 19 pieces of 12 cm.
Three traffic lights change every 48 s, 72 s and 108 s. If they change together at 8:00:00 a.m., when will they next change together?
Solution: LCM = 2⁴ × 3³ = 432 s = 7 min 12 s. Next together at 8:07:12 a.m.
Find the smallest number that leaves remainder 3 when divided by 6, 8 and 12.
Solution: LCM(6, 8, 12) = 24. The number is 24 + 3 = 27.
A room is 6 m 24 cm long and 4 m 32 cm wide. Find the largest square tile that fits exactly, and how many tiles are needed.
Solution: 624 cm and 432 cm. HCF: 624 = 2⁴ × 3 × 13, 432 = 2⁴ × 3³ → HCF = 48 cm. Tiles = (624 ÷ 48) × (432 ÷ 48) = 13 × 9 = 117 tiles.
9. Master Formula Table
| Idea | Rule |
|---|---|
| HCF (prime method) | common primes, smaller powers |
| LCM (prime method) | all primes, larger powers |
| Two numbers | HCF × LCM = a × b |
| Co-prime numbers | HCF = 1, LCM = a × b |
| Euclid's step | HCF(a, b) = HCF(b, a − b) |
| Same remainder r | LCM + r |
| HCF ≤ smaller number | LCM ≥ larger number |
10. Master Summary Table
| Topic | Key idea | Exam focus |
|---|---|---|
| HCF | Greatest common factor | Largest equal parts |
| LCM | Least common multiple | Next meeting time |
| Methods | Listing, primes, division | Choose efficiently |
| Relation | HCF × LCM = product | Find a missing number |
| Word problems | Key words decide | HCF vs LCM |
11. Mind Map
- Common factors
- Smaller powers
- Euclid's method
- Common multiples
- Larger powers
- Common division
- HCF × LCM = ab
- Co-primes
- Cutting, tiling → HCF
- Bells, lights → LCM
12. Tips and Tricks
- Always check: HCF divides both numbers; LCM is divisible by both numbers.
- Convert all measurements to the same unit before finding HCF or LCM.
- If one number divides the other (like 8 and 24), HCF = smaller, LCM = larger.
- Using HCF for "when will they meet again" problems (it needs LCM).
- Applying HCF × LCM = product to three numbers.
- Taking larger powers for the HCF.
13. Important One-liners
- HCF is the greatest number dividing all the given numbers.
- LCM is the smallest number divisible by all the given numbers.
- HCF of co-prime numbers is 1.
- LCM of co-prime numbers is their product.
- For two numbers, HCF × LCM = product of the numbers.
- The HCF divides the LCM.
- HCF(a, b) also divides a − b.
14. Exam-pattern Questions with Detailed Answers
Find the HCF of 18 and 27.
- 3
- 6
- 9
- 54
Show answer
18 = 2 × 3², 27 = 3³. Common: 3² = 9.
Answer: (C) 9
Find the LCM of 15 and 20.
Show answer
15 = 3 × 5, 20 = 2² × 5. LCM = 2² × 3 × 5 = 60.
Answer: 60
What is the LCM of 7 and 11?
Show answer
They are co-prime, so LCM = 77.
Answer: 77
Find HCF and LCM of 90, 120 and 150.
Show answer
90 = 2 × 3² × 5; 120 = 2³ × 3 × 5; 150 = 2 × 3 × 5². HCF = 2 × 3 × 5 = 30. LCM = 2³ × 3² × 5² = 1800.
Answer: HCF 30, LCM 1800
Two numbers have HCF 8 and product 768. Find their LCM.
Show answer
LCM = product ÷ HCF = 768 ÷ 8 = 96.
Answer: 96
Can two numbers have HCF 12 and LCM 90?
- Yes
- No
Show answer
The HCF must divide the LCM. 90 ÷ 12 is not a whole number.
Answer: (B) No
Use Euclid's method to find HCF(391, 323).
Show answer
391 = 1 × 323 + 68; 323 = 4 × 68 + 51; 68 = 1 × 51 + 17; 51 = 3 × 17 + 0. HCF = 17.
Answer: 17
Sweets: 120 laddoos and 168 barfis are packed in boxes, each box with the same number of one kind only, as large as possible. How many sweets per box?
Show answer
HCF(120, 168): 120 = 2³ × 3 × 5, 168 = 2³ × 3 × 7 → 24.
Answer: 24 sweets per box
Find the smallest 4-digit number divisible by 12, 15 and 20.
Show answer
LCM(12, 15, 20) = 60. Smallest 4-digit multiple of 60: 1000 ÷ 60 = 16.67, so 17 × 60 = 1020.
Answer: 1020
Three runners take 30 s, 45 s and 60 s to complete a round of a track. Starting together, after how many minutes will they all be at the start again?
Show answer
LCM(30, 45, 60) = 180 s = 3 minutes.
Answer: 3 minutes
Find the greatest number that divides 245 and 1029 leaving remainder 5 in each case.
Show answer
The number exactly divides 240 and 1024. 240 = 2⁴ × 3 × 5, 1024 = 2¹⁰. HCF = 2⁴ = 16.
Answer: 16
The LCM of two co-prime numbers is 391 and one of them is 17. Find the other and their HCF.
Show answer
For co-primes, LCM = product, so the other = 391 ÷ 17 = 23. HCF = 1.
Answer: 23; HCF 1
15. Quick Revision Summary
- HCF: common primes with smaller powers; LCM: all primes with larger powers.
- Euclid: HCF(a, b) = HCF(b, remainder) — fast for big numbers.
- HCF × LCM = a × b for two numbers; HCF always divides LCM.
- Cutting/grouping/tiling → HCF; meeting again/smallest common amount → LCM.