HCF (Highest Common Factor) and LCM (Lowest Common Multiple) are two of the most tested topics in Class 10 CBSE Maths, SSC exams, and competitive tests like JEE, CUET, and banking exams. They appear in Chapter 1 (Real Numbers) of the NCERT Class 10 syllabus and are the foundation for understanding fractions, ratios, and number theory.
This guide covers every method to calculate HCF and LCM with step-by-step examples — from prime factorisation to the Euclidean Algorithm — plus the key relationship between them, shortcut tricks, and exam-type problems.
What Is HCF?
HCF (Highest Common Factor) — also called GCD (Greatest Common Divisor) — is the largest number that divides two or more numbers exactly (with no remainder).
Example: Find HCF of 12 and 18.
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
- Common factors: 1, 2, 3, 6
- HCF = 6 (the largest common factor)
What Is LCM?
LCM (Lowest Common Multiple) is the smallest positive number that is a multiple of two or more numbers.
Example: Find LCM of 4 and 6.
- Multiples of 4: 4, 8, 12, 16, 20, 24...
- Multiples of 6: 6, 12, 18, 24...
- LCM = 12 (the smallest common multiple)
The Key Relationship: HCF × LCM = Product of Two Numbers
For any two numbers a and b:
HCF(a, b) × LCM(a, b) = a × b
This is one of the most important formulas in CBSE Class 10 and appears directly in board exams.
Example: Two numbers are 12 and 18.
- HCF = 6, Product = 12 × 18 = 216
- LCM = 216 / 6 = 36 ✓ (verify: multiples of 36 include both 12 and 18? 36/12=3 ✓, 36/18=2 ✓)
Warning: This relationship holds only for two numbers. It does NOT extend to three or more numbers.
Method 1: Prime Factorisation
This is the most conceptually clear method and is required by CBSE board exams.
How to find HCF using Prime Factorisation
Step 1: Write the prime factorisation of each number. Step 2: Identify common prime factors. Step 3: HCF = product of common prime factors, each raised to the lowest power.
Example: Find HCF of 36 and 84
- 36 = 2² × 3²
- 84 = 2² × 3 × 7
Common prime factors: 2 (lowest power = 2²) and 3 (lowest power = 3¹)
HCF = 2² × 3 = 4 × 3 = 12
How to find LCM using Prime Factorisation
Step 1: Write the prime factorisation of each number. Step 2: LCM = product of all prime factors, each raised to the highest power.
Example: Find LCM of 36 and 84
- 36 = 2² × 3²
- 84 = 2² × 3 × 7
All prime factors with highest powers: 2² × 3² × 7
LCM = 4 × 9 × 7 = 252
Verification: HCF × LCM = 12 × 252 = 3024. Product of numbers = 36 × 84 = 3024 ✓
Example 2: Three Numbers (CBSE-style)
Find HCF and LCM of 12, 15, and 21.
Prime factorisations:
- 12 = 2² × 3
- 15 = 3 × 5
- 21 = 3 × 7
HCF = only 3 appears in all three → HCF = 3
LCM = 2² × 3 × 5 × 7 = 4 × 3 × 5 × 7 = 420
Method 2: Division Method (Long Division / Euclidean Algorithm)
This method is faster for large numbers and is the basis for the Euclidean Algorithm — one of the oldest algorithms in mathematics.
Euclidean Algorithm for HCF
Rule: HCF(a, b) = HCF(b, a mod b) — keep dividing until remainder = 0.
Example: Find HCF of 48 and 18
Step 1: 48 ÷ 18 = 2 remainder 12
Step 2: 18 ÷ 12 = 1 remainder 6
Step 3: 12 ÷ 6 = 2 remainder 0 ← STOP
HCF = 6 (the last non-zero divisor)
Example: Find HCF of 252 and 105
Step 1: 252 ÷ 105 = 2 remainder 42
Step 2: 105 ÷ 42 = 2 remainder 21
Step 3: 42 ÷ 21 = 2 remainder 0 ← STOP
HCF = 21
Finding HCF of Three Numbers via Division Method
Find HCF(a, b, c) = HCF(HCF(a, b), c)
Example: HCF(36, 24, 60)
-
HCF(36, 24):
- 36 ÷ 24 = 1 rem 12
- 24 ÷ 12 = 2 rem 0 → HCF = 12
-
HCF(12, 60):
- 60 ÷ 12 = 5 rem 0 → HCF = 12
HCF(36, 24, 60) = 12
Method 3: Listing Method (For Small Numbers)
Suitable for numbers under 30 in competitive exam MCQs.
Find HCF of 24 and 36:
- 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
- HCF = 12
Method 4: Common Division Method (Ladder Method for LCM)
This is the fastest method for finding LCM of 2–4 numbers and is widely used in competitive exams.
Example: LCM of 8, 12, and 18
Divide all numbers by smallest prime that divides at least one of them:
2 | 8 12 18
2 | 4 6 9
2 | 2 3 9
3 | 1 3 9
3 | 1 1 3
| 1 1 1
LCM = 2 × 2 × 2 × 3 × 3 = 72
Worked Examples (NCERT & Board Level)
Example 1 (NCERT 1.2)
Find HCF and LCM of 26 and 91 using prime factorisation.
- 26 = 2 × 13
- 91 = 7 × 13
HCF = 13 (only common prime factor) LCM = 2 × 7 × 13 = 182
Verify: 13 × 182 = 2366. 26 × 91 = 2366 ✓
Example 2 (NCERT 1.2)
Find LCM and HCF of 6, 72, and 120 using prime factorisation.
- 6 = 2 × 3
- 72 = 2³ × 3²
- 120 = 2³ × 3 × 5
HCF = 2¹ × 3¹ = 6 (minimum powers of common factors) LCM = 2³ × 3² × 5 = 8 × 9 × 5 = 360 (maximum powers of all factors)
Example 3 (Application Problem)
Two bells ring every 15 minutes and 20 minutes respectively. If they ring together at 8:00 AM, when will they next ring together?
They ring together at the LCM of 15 and 20.
- 15 = 3 × 5
- 20 = 2² × 5
LCM = 2² × 3 × 5 = 60 minutes
They will ring together again at 9:00 AM.
Example 4 (Application Problem)
Three tanks can be filled in 8, 12, and 16 hours respectively. All three pipes are opened together. After how many hours will all three be simultaneously full?
LCM(8, 12, 16):
- 8 = 2³
- 12 = 2² × 3
- 16 = 2⁴
LCM = 2⁴ × 3 = 48 hours
Example 5 (NCERT 1.2, Q5)
Check whether 6^n can end with digit 0, for any natural number n.
A number ends in 0 only if it has 2 and 5 as prime factors. 6^n = (2 × 3)^n = 2^n × 3^n — this has only 2 and 3 as prime factors, never 5.
Therefore, 6^n cannot end with digit 0 for any natural number n.
(This is a classic CBSE proof problem using Fundamental Theorem of Arithmetic — always appears in board exams.)
Quick Shortcuts for Competitive Exams
Shortcut 1: HCF of numbers that share a common factor
If two numbers are ka and kb (share factor k), then:
- HCF(ka, kb) = k × HCF(a, b)
Example: HCF(14, 21) = 7 × HCF(2, 3) = 7 × 1 = 7
Shortcut 2: LCM when one number is a multiple of another
If b = ka (one number is a multiple of the other):
- LCM(a, b) = b (the larger number)
- HCF(a, b) = a (the smaller number)
Example: LCM(6, 18) = 18, HCF(6, 18) = 6
Shortcut 3: Co-prime numbers
Two numbers are co-prime if HCF = 1. For co-prime numbers:
- LCM = a × b
Example: HCF(9, 14) = 1 (co-prime) → LCM = 9 × 14 = 126
Co-prime pairs to memorise: (8, 9), (4, 9), (5, 6), (5, 7), (3, 7), (11, 13)
Shortcut 4: Finding one number when HCF, LCM, and the other number are given
Using HCF × LCM = a × b:
Second number = (HCF × LCM) / First number
Example: HCF = 4, LCM = 60, one number = 12. Find the other. Other = (4 × 60) / 12 = 240 / 12 = 20
Shortcut 5: LCM of fractions and HCF of fractions
LCM of fractions = LCM of numerators / HCF of denominators HCF of fractions = HCF of numerators / LCM of denominators
Example: LCM(2/3, 4/9, 5/6)
- LCM of numerators: LCM(2, 4, 5) = 20
- HCF of denominators: HCF(3, 9, 6) = 3
- Answer: 20/3
Example: HCF(2/3, 4/9, 5/6)
- HCF of numerators: HCF(2, 4, 5) = 1
- LCM of denominators: LCM(3, 9, 6) = 18
- Answer: 1/18
Common Exam Mistakes to Avoid
Mistake 1: Confusing HCF and LCM in word problems
HCF → problems about dividing into equal groups, finding maximum size, distributing items equally LCM → problems about finding when events coincide again (bells, buses, blinking lights), minimum quantity
Memory trick:
- HCF → Highest → splitting, maximum common measure
- LCM → Lowest → combining, minimum time to sync
Mistake 2: Applying HCF × LCM = a × b to three numbers
This formula works only for two numbers. For three or more numbers, there is no such direct formula.
Mistake 3: Wrong prime factorisation
Always verify by multiplying back.
2 × 2 × 3 × 7 = 84 ✓ Not 2 × 4 × 3 × 7 = 168 ✗ (4 is not prime)
Mistake 4: Missing prime factors in LCM
In LCM, include all prime factors from all numbers (each at its highest power). Students often forget factors that appear in only one of the numbers.
HCF includes only factors common to ALL numbers. LCM includes ALL factors from ANY number.
HCF and LCM in CBSE Board Exam Questions
Chapter 1 (Real Numbers) — Typical Board Question Types
Type 1 — Direct calculation: "Find HCF and LCM of 84 and 90 using prime factorisation."
Type 2 — Verification: "Verify that LCM × HCF = product of two numbers for 26 and 91."
Type 3 — Application (tiles/containers/bells): "What is the largest tile size that can exactly tile a 1260 cm × 840 cm floor?" → Find HCF(1260, 840) = 420 cm
Type 4 — Given HCF, find LCM or vice versa: "The HCF of two numbers is 9 and LCM is 360. If one number is 45, find the other."
Type 5 — Proof using FTA: "Prove that 3 + 2√5 is irrational." (Related topic — uses uniqueness of prime factorisation)
Topic Map: Where HCF/LCM Connects to Other CBSE Topics
| Topic | Connection to HCF/LCM |
|---|---|
| Fractions | Simplifying fractions uses HCF of numerator and denominator |
| Adding fractions | Finding common denominator uses LCM |
| Ratio & proportion | HCF simplifies ratios to lowest terms |
| Euclidean Algorithm | Basis of modern cryptography (RSA) |
| Real Numbers | Fundamental Theorem of Arithmetic: every integer has a unique prime factorisation |
| Polynomials | GCD/LCM of polynomial expressions (Class 10 extension) |
Practice Problems (Board + Competitive Level)
Easy (CBSE):
- Find HCF and LCM of 48 and 72.
- If HCF of two numbers is 6 and LCM is 180, and one number is 30, find the other.
- Find LCM(4, 6, 9, 12).
Medium (SSC/Banking): 4. Three buses leave a bus stop every 15, 18, and 24 minutes. If they all leave together at 8:00 AM, at what time do they leave together again? 5. Find the largest number that divides 2053 and 967 leaving remainders 5 and 7 respectively. 6. Find HCF(a²b, ab²) given a and b are prime numbers.
Hard (NCERT Exemplar/JEE): 7. Show that every positive even integer is of the form 2m and every odd integer is of the form 2m+1 for some integer m. 8. Prove that √3 is irrational using the Fundamental Theorem of Arithmetic.
Answers:
- HCF = 24, LCM = 144
- Other number = 36
- LCM = 36
- LCM(15, 18, 24) = 360 minutes = 6 hours → 2:00 PM
- Numbers leave remainder 5 and 7 → Subtract: (2053−5)=2048 and (967−7)=960. HCF(2048, 960) = 64 → Largest number = 64
- HCF = ab (lowest power of each), LCM = a²b² (highest power of each)
Frequently Asked Questions
What is the difference between HCF and GCD?
They are the same thing. HCF (Highest Common Factor) is the term used in Indian textbooks (CBSE/ICSE). GCD (Greatest Common Divisor) is the term used in international curricula (GCSE, IB, JEE). Both mean the same: the largest number that divides two or more numbers without leaving a remainder.
Which method is fastest for HCF in competitive exams?
The Euclidean Algorithm (Division Method) is fastest for two large numbers. The Ladder Method is fastest for LCM of multiple small numbers. For very small numbers (under 20), the listing method is quickest mentally.
Can HCF be larger than LCM?
No. HCF ≤ min(a, b) and LCM ≥ max(a, b). The HCF is always less than or equal to both numbers; the LCM is always greater than or equal to both numbers.
What is HCF of two consecutive numbers?
Always 1. Consecutive numbers (e.g., 7 and 8, 99 and 100) are always co-prime — they share no common factors other than 1.
What is LCM of two consecutive numbers?
Their product. Since consecutive numbers are co-prime (HCF=1), LCM = a × b.
How are HCF and LCM used in real life?
- Scheduling: Finding when two recurring events coincide (bus routes, machine maintenance schedules)
- Dividing equally: Splitting items into equal groups without remainder
- Music: Finding the beat alignment of two rhythms with different periods
- Construction: Finding the largest tile that fits a room exactly (no cutting)
- Cryptography: The Euclidean Algorithm is at the heart of RSA encryption (used for every HTTPS connection)
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