LCM and HCF are two of the most essential mathematical foundations taught in Class 6. LCM stands for Least Common Multiple, and HCF stands for Highest Common Factor. This complete revision guide covers formal definitions, prime factor Venn diagrams, Euclidean division, ladder methods, solved word problems, common exam traps, and practice exercises.
The HCF (Highest Common Factor) of two or more numbers is the largest number that divides all of them exactly without leaving any remainder. In international curricula, it is also referred to as the GCD (Greatest Common Divisor) or GCF (Greatest Common Factor).
The LCM (Least Common Multiple) of two or more numbers is the smallest positive integer that is exactly divisible by all of them without leaving a remainder.
Understanding the relationship between factors, primes, and multiples visually makes Class 6 maths intuitive and permanent.
Prime factor circles for 60 (2 × 2 × 3 × 5) and 72 (2 × 2 × 2 × 3 × 3). The intersection (middle overlap) contains the common prime factors: 2 × 2 × 3 = 12 (HCF). Multiplying all factors across the entire union gives the LCM: 5 × (12) × 6 = 360.
How to know whether a word problem requires HCF or LCM in exams. Follow the keyword clues:
There are three standard methods to find the HCF of numbers in the Class 6 syllabus.
List all positive factors of each number, locate common factors, and choose the greatest one.
Express each number as a product of prime powers and multiply common primes with smallest exponent.
Continuously divide divisor by remainder until remainder is 0. The last divisor is the HCF.
Steps:
Example: Find HCF of 24 and 36
Steps:
Example: Find HCF of 60 and 72
The long division method is the quickest method for large numbers without having to list factors.
Divide 48 by 18, then divide 18 by the remainder 12, then divide 12 by the remainder 6. When remainder becomes 0, the last divisor (6) is the HCF.
Class 6 students learn three methods to find the Least Common Multiple.
List multiples of each number, locate common multiples, and select the least one.
Decompose numbers into prime powers and multiply the highest exponent of every prime.
Divide numbers simultaneously by common prime divisors until no two share any factor.
Steps:
Example: Find LCM of 6 and 8
Steps:
Example: Find LCM of 36 and 60
The ladder method is the most practical exam method for finding LCM of two or three numbers at once.
Example: Find LCM of 21, 35 and 42
| Prime Divisor | 21 | 35 | 42 | Operation Note |
|---|---|---|---|---|
| 7 | 3 | 5 | 6 | 7 divides 21, 35, and 42 |
| 3 | 1 | 5 | 2 | 3 divides 3 and 6 (carry forward 5) |
| Bottom Row | 1 | 5 | 2 | Remaining numbers are pairwise coprime |
For any two positive integers, there is a fundamental law connecting their HCF and LCM:
Question: The HCF of 18 and 24 is 6. Find their LCM using the formula.
Question: The LCM of two numbers is 30 and their HCF is 5. If one of the numbers is 10, find the other number.
Step-by-step solutions to classical Class 6 NCERT & school board exam problems.
Question: Renu purchases two bags of fertiliser weighing 75 kg and 69 kg. Find the maximum value of weight that can measure the fertiliser weight an exact number of times.
Question: Three boys step off together from the same spot. Their steps measure 63 cm, 70 cm and 77 cm. What is the minimum distance each should cover so that all can cover the distance in complete steps?
Question: The length, breadth and height of a room are 825 cm, 675 cm and 450 cm respectively. Determine the longest tape which can measure the three dimensions of the room exactly.
Question: The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they change simultaneously at 7:00:00 AM, at what time will they change together again?
Question: Determine the smallest 3-digit number exactly divisible by 6, 8 and 12.
Examiners often test these specific misconceptions. Keep them in mind during exams.
| ❌ Common Mistake | Why it's wrong | ✅ Correct Mathematical Fact |
|---|---|---|
| Confusing HCF and LCM values | LCM is a multiple (large), HCF is a factor (small) | HCF ≤ smallest input; LCM ≥ largest input |
| Writing HCF as 0 for co-prime numbers | 0 is never a factor of any non-zero number | The HCF of co-prime numbers is always 1 |
| Assuming common divisibility means product divisibility | A number divisible by 4 and 6 is divisible by 12, not necessarily 24 | Always divide by LCM(a, b), not a × b |
| Using HCF × LCM = Product for 3 numbers | The formula fails when three numbers share pairwise common factors | Use prime factorization or division method for 3+ numbers |
| Mixing up word problem keywords | Guessing the operation without checking whether the answer should be larger or smaller | "Maximum / Divide" → HCF; "Simultaneous / Minimum" → LCM |
A complete side-by-side reference for Class 6 revision.
| Aspect | HCF (Highest Common Factor) | LCM (Least Common Multiple) |
|---|---|---|
| Full Name | Highest Common Factor / GCD / GCF | Least Common Multiple / LCD |
| Meaning | Largest number that divides all given numbers | Smallest number that is divisible by all given numbers |
| Relative Size | Always ≤ smallest given number | Always ≥ largest given number |
| Prime Factor Rule | Take common prime factors with lowest power | Take every prime factor with highest power |
| Division Method | Euclidean long division until remainder = 0 | Ladder table until bottom numbers are co-prime |
| Co-Prime Numbers | HCF is always 1 | LCM is always the product of the numbers |
| Real-world Role | Splitting into equal groups, cutting equal lengths | Matching cycles, synchronized bells & lights, fraction LCD |
| Connecting Law | For two numbers A and B: HCF(A, B) × LCM(A, B) = A × B | |
Quick answers to common student questions about Class 6 LCM and HCF.
HCF (Highest Common Factor) is the largest number that divides two or more given numbers exactly without leaving any remainder. It is also called GCD (Greatest Common Divisor). For example, HCF of 12 and 18 is 6.
LCM (Least Common Multiple) is the smallest positive number that is exactly divisible by two or more given numbers. For example, LCM of 4 and 6 is 12.
For two numbers: HCF × LCM = Product of the two numbers. This formula is very useful for finding the unknown number or LCM when HCF is known. It does not apply to sets of three numbers.
The HCF of co-prime numbers is always 1, and their LCM is their product. For example, 3 and 5 are co-prime: HCF = 1, LCM = 15.
If the problem asks for the greatest, maximum, largest, or longest quantity, use HCF. If it asks for the least, minimum, smallest, or simultaneous repetition, use LCM.
No. For any positive integers, HCF is always ≤ the smallest number, and LCM is always ≥ the largest number. Therefore, HCF ≤ LCM always.
Test your understanding with these practice exercises.
Essential definitions for Class 6 examinations.