LCM AND HCF
Topic: Least Common Multiple (LCM) and Highest Common Factor (HCF) Subject: Quantitative Aptitude / Mathematics Useful for: JKSSB, SSC, Banking, Railway, Patwari, Police, CTET, TET and other competitive examinations.
1\. INTRODUCTION
LCM and HCF are important topics in the Mathematics and Quantitative Aptitude sections of competitive examinations. Questions based on these topics are frequently asked in government recruitment examinations.
Both concepts are related to factors and multiples of numbers. A proper understanding of LCM and HCF helps candidates solve mathematical problems quickly and accurately.
There are two main concepts:
* LCM – Least Common Multiple
* HCF – Highest Common Factor
### 2\. WHAT IS HCF?
HCF stands for Highest Common Factor.
It is the greatest number that divides two or more given numbers exactly without leaving any remainder.
HCF is also known as:
* GCD – Greatest Common Divisor.
* GCM – Greatest Common Measure.
Example: Find the 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
The greatest common factor is 6.
Therefore, HCF = 6.
### 3\. WHAT IS LCM?
LCM stands for Least Common Multiple.
It is the smallest positive number that is exactly divisible by all the given numbers.
Example: Find the LCM of 4 and 6.
Multiples of 4:
4, 8, 12, 16, 20, 24, 28, 32...
Multiples of 6:
6, 12, 18, 24, 30, 36...
Common multiples:
12, 24, 36...
The smallest common multiple is 12.
Therefore, LCM = 12.
### 4\. DIFFERENCE BETWEEN LCM AND HCF
| HCF | LCM |
| ----------------------------------- | ---------------------------------- |
| Highest Common Factor | Least Common Multiple |
| Greatest common divisor | Smallest common multiple |
| Deals with factors | Deals with multiples |
| Used for maximum equal distribution | Used for minimum common occurrence |
| Generally smaller | Generally larger |
| Also called GCD | Also called Least Common Multiple |
Important:
HCF cannot be greater than the smallest given positive integer, while LCM cannot be smaller than the greatest given positive integer.
### 5\. IMPORTANT MATHEMATICAL TERMS
1\. Factor: A number that divides another number exactly.
Example: Factors of 20 are 1, 2, 4, 5, 10 and 20.
2\. Multiple: A number obtained by multiplying a given number by a positive integer.
Example: Multiples of 5 are 5, 10, 15, 20, 25 and 30.
3\. Prime Number: A number greater than 1 having exactly two factors: 1 and itself.
Examples: 2, 3, 5, 7, 11, 13, 17 and 19.
4\. Composite Number: A number greater than 1 having more than two factors.
Examples: 4, 6, 8, 9, 10, 12 and 15.
5\. Co-prime Numbers: Two numbers whose HCF is 1.
Examples: 8 and 15, 9 and 16, 14 and 25.
6\. Prime Factorisation: Expressing a number as a product of prime factors.
Example:
60 = 2 × 2 × 3 × 5
60 = 2² × 3 × 5
### 6\. METHODS OF FINDING HCF
There are three important methods of finding HCF.
### Method 1: Factor Method
In this method, we write all the factors of the given numbers and identify their greatest common factor.
Example: Find the 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.
The greatest common factor is 12.
Answer: HCF = 12.
### Method 2: Prime Factorisation Method
In this method, we express all given numbers as products of prime factors.
Example: Find the HCF of 48 and 60.
Prime factorisation:
48 = 2 × 2 × 2 × 2 × 3
48 = 2⁴ × 3
60 = 2 × 2 × 3 × 5
60 = 2² × 3 × 5
Common prime factors with the smallest powers:
HCF = 2² × 3
HCF = 4 × 3
Answer: HCF = 12.
IMPORTANT RULE:
For HCF, take only the common prime factors with their smallest powers.
### Method 3: Division Method (Euclidean Algorithm)
This is one of the quickest methods of finding HCF, particularly when large numbers are given.
Example: Find the HCF of 144 and 96.
Step 1:
144 ÷ 96 = 1, remainder 48.
Step 2:
96 ÷ 48 = 2, remainder 0.
Since the remainder becomes zero, the last non-zero divisor is the HCF.
Answer: HCF = 48.
Division process:
144 = 96 × 1 + 48
96 = 48 × 2 + 0
IMPORTANT RULE:
Continue dividing the previous divisor by the remainder until the remainder becomes zero. The last non-zero divisor is the HCF.
### 7\. METHODS OF FINDING LCM
There are three commonly used methods of finding LCM.
### Method 1: Listing Multiples
Write the multiples of each number and identify the smallest common multiple.
Example: Find the LCM of 8 and 12.
Multiples of 8:
8, 16, 24, 32, 40, 48...
Multiples of 12:
12, 24, 36, 48, 60...
Common multiples:
24, 48, 72...
The smallest common multiple is 24.
Answer: LCM = 24.
### Method 2: Prime Factorisation Method
Express all numbers as products of prime factors and select every prime factor with its highest power.
Example: Find the LCM of 24 and 36.
Prime factorisation:
24 = 2³ × 3
36 = 2² × 3²
Take the highest powers:
LCM = 2³ × 3²
LCM = 8 × 9
Answer: LCM = 72.
IMPORTANT RULE:
For LCM, take all prime factors with their greatest powers.
### Method 3: Division Method
This method is useful when three or more numbers are given.
Example: Find the LCM of 12, 18 and 24.
```text
2 | 12 18 24
2 | 6 9 12
2 | 3 9 6
3 | 3 9 3
3 | 1 3 1
| 1 1 1
```
Multiply all divisors:
LCM = 2 × 2 × 2 × 3 × 3
LCM = 8 × 9
Answer: LCM = 72.
### 8\. IMPORTANT RELATIONSHIP BETWEEN LCM AND HCF
For any two positive integers:
LCM × HCF = First Number × Second Number
This is one of the most important formulas for competitive examinations.
Example 1:
Find the LCM of 18 and 24 if their HCF is 6.
Given:
First number = 18
Second number = 24
HCF = 6
Formula:
LCM × HCF = First Number × Second Number
LCM × 6 = 18 × 24
LCM = (18 × 24) ÷ 6
LCM = 432 ÷ 6
Answer: LCM = 72.
Example 2:
Find the HCF of two numbers if their product is 2160 and LCM is 180.
Formula:
HCF × LCM = Product of two numbers
HCF × 180 = 2160
HCF = 2160 ÷ 180
Answer: HCF = 12.
IMPORTANT NOTE:
The product relationship applies directly to two positive integers, not generally to three or more integers.
### 9\. IMPORTANT PROPERTIES OF LCM AND HCF
* HCF of two co-prime numbers is always 1.
* LCM of two co-prime numbers is equal to their product.
* HCF of two equal numbers is the number itself.
* LCM of two equal numbers is the number itself.
* HCF of 1 and any positive integer is 1.
* LCM of 1 and any positive integer is that integer.
* HCF is always a factor of every given number.
* LCM is always a multiple of every given number.
* HCF of two numbers cannot be greater than the smaller number.
* LCM of two numbers cannot be smaller than the greater number.
* If one number divides another exactly, the smaller number is the HCF and the greater number is the LCM.
* HCF of two numbers always divides their difference.
### Special Cases
| Numbers | HCF | LCM |
| ------- | --- | --- |
| 5, 7 | 1 | 35 |
| 8, 16 | 8 | 16 |
| 12, 12 | 12 | 12 |
| 1, 25 | 1 | 25 |
| 9, 15 | 3 | 45 |
| 18, 30 | 6 | 90 |
### 10\. HCF OF FRACTIONS
Formula:
HCF of Fractions = HCF of Numerators ÷ LCM of Denominators.
Example: Find the HCF of 2/3, 4/9 and 8/15.
Step 1: Find the HCF of numerators.
HCF of 2, 4 and 8 = 2.
Step 2: Find the LCM of denominators.
LCM of 3, 9 and 15 = 45.
Therefore:
HCF = 2/45.
Answer: 2/45.
### 11\. LCM OF FRACTIONS
Formula:
LCM of Fractions = LCM of Numerators ÷ HCF of Denominators.
Example: Find the LCM of 2/3, 4/9 and 8/15.
Step 1: LCM of numerators (2, 4, 8) = 8.
Step 2: HCF of denominators (3, 9, 15) = 3.
Therefore:
LCM = 8/3.
Answer: 8/3.
IMPORTANT:
These standard fraction formulas are used when the fractions are expressed in their lowest terms.
### 12\. HCF OF DECIMAL NUMBERS
To calculate HCF of decimal numbers, follow these steps:
* Make the number of decimal places equal by adding zeros if necessary.
* Remove the decimal points.
* Calculate the HCF of the resulting integers.
* Divide the answer by the corresponding power of 10.
Example: Find the HCF of 1.2 and 1.8.
Step 1:
1.2 and 1.8 have one decimal place.
Step 2:
Remove decimal points.
12 and 18.
Step 3:
HCF of 12 and 18 = 6.
Step 4:
Divide by 10.
HCF = 6 ÷ 10 = 0.6.
Answer: 0.6.
### 13\. LCM OF DECIMAL NUMBERS
Follow the same process, but calculate the LCM after removing decimal points.
Example: Find the LCM of 1.2 and 1.8.
Step 1:
Remove decimal points.
12 and 18.
Step 2:
LCM of 12 and 18 = 36.
Step 3:
Divide by 10.
LCM = 36 ÷ 10 = 3.6.
Answer: 3.6.
### 14\. APPLICATIONS OF HCF IN WORD PROBLEMS
HCF is generally used in problems involving:
* Equal distribution of objects.
* Making the largest possible groups.
* Cutting objects into equal pieces.
* Dividing land into maximum-sized square plots.
* Finding the greatest possible measurement.
* Arranging items into identical groups without leftovers.
### Type 1: Equal Distribution
Question:
A teacher has 48 pencils and 60 pens. She wants to distribute them equally among the maximum number of students so that nothing remains. Find the maximum number of students.
Solution:
HCF of 48 and 60 = 12.
Therefore, the maximum number of students is 12.
Each student will receive:
Pencils = 48 ÷ 12 = 4.
Pens = 60 ÷ 12 = 5.
Answer: 12 students.
### Type 2: Cutting Ropes into Equal Pieces
Question:
Three ropes have lengths of 24 metres, 36 metres and 60 metres. They are to be cut into equal pieces of maximum possible length. Find the length of each piece.
Solution:
Required length = HCF of 24, 36 and 60.
HCF = 12.
Answer: 12 metres.
### Type 3: Largest Square Tile
Question:
A rectangular floor is 240 cm long and 180 cm wide. Find the largest possible square tile that can cover the floor without cutting any tile.
Solution:
Required tile size = HCF of 240 and 180.
HCF = 60.
Therefore, the side of each square tile is 60 cm.
Number of tiles:
= (240 ÷ 60) × (180 ÷ 60)
= 4 × 3
Answer: 12 tiles.
### 15\. APPLICATIONS OF LCM IN WORD PROBLEMS
LCM is commonly used in problems involving:
* Repeated events.
* Bells ringing at different intervals.
* Traffic lights changing at different intervals.
* People visiting a place at regular intervals.
* Machines operating periodically.
* Finding a common denominator.
* Finding the smallest number divisible by given numbers.
### Type 1: Bells Ringing Together
Question:
Three bells ring at intervals of 12, 15 and 20 seconds. If they ring together at 10:00 AM, when will they ring together again?
Solution:
LCM of 12, 15 and 20.
12 = 2² × 3
15 = 3 × 5
20 = 2² × 5
LCM = 2² × 3 × 5
LCM = 60 seconds.
Therefore, they will ring together again after 60 seconds.
Answer: 10:01 AM.
### Type 2: Traffic Lights
Question:
Three traffic lights change after every 30, 45 and 60 seconds. If they change together now, after how many seconds will they change together again?
Solution:
LCM of 30, 45 and 60 = 180.
Answer: 180 seconds.
### Type 3: Repeated Visits
Question:
A person visits a library every 4 days, another every 6 days and a third every 8 days. If all three visit today, after how many days will they visit together again?
Solution:
LCM of 4, 6 and 8 = 24.
Answer: 24 days.
### 16\. IMPORTANT SHORTCUTS AND TRICKS
Trick 1: When Two Numbers Are Given
LCM = (Product of Numbers) ÷ HCF.
Example:
Numbers = 36 and 48.
HCF = 12.
LCM = (36 × 48) ÷ 12.
LCM = 144.
Trick 2: When LCM and HCF Are Given
Product of Two Numbers = LCM × HCF.
Example:
LCM = 120 and HCF = 5.
Product = 120 × 5 = 600.
Trick 3: Co-prime Numbers
If two numbers are co-prime:
HCF = 1.
LCM = Product of Numbers.
Example:
17 and 19.
LCM = 17 × 19 = 323.
Trick 4: When One Number Is a Multiple of Another
If one number is completely divisible by another:
HCF = Smaller Number.
LCM = Greater Number.
Example:
16 and 64.
HCF = 16.
LCM = 64.
Trick 5: Consecutive Numbers
Two consecutive positive integers are always co-prime.
Example:
21 and 22.
HCF = 1.
LCM = 21 × 22 = 462.
Trick 6: Difference Method
The HCF of two numbers always divides their difference.
Example:
Find the HCF of 84 and 126.
Difference = 126 − 84 = 42.
126 ÷ 84 = remainder 42.
84 ÷ 42 = remainder 0.
HCF = 42.
Trick 7: Finding the Number of Tiles or Pieces
Side of Square = HCF of Length and Breadth.
Number of Squares = (Length ÷ Side) × (Breadth ÷ Side).
Trick 8: Finding the Next Common Time
When two or more events repeat after fixed intervals:
Time of Next Meeting = LCM of Given Intervals.
### 17\. FINDING TWO NUMBERS WHEN THEIR HCF AND LCM ARE GIVEN
This is another important question type.
Question:
The HCF of two numbers is 6 and their LCM is 180. If one number is 30, find the other number.
Solution:
LCM × HCF = First Number × Second Number.
180 × 6 = 30 × Second Number.
Second Number = (180 × 6) ÷ 30.
Second Number = 36.
Answer: 36.
### 18\. FINDING THE GREATEST NUMBER THAT DIVIDES SEVERAL NUMBERS
Question:
Find the greatest number that divides 45, 75 and 105 exactly.
Solution:
Required number = HCF of 45, 75 and 105.
45 = 3² × 5.
75 = 3 × 5².
105 = 3 × 5 × 7.
Common prime factors:
3 × 5 = 15.
Answer: 15.
### 19\. FINDING THE LEAST NUMBER DIVISIBLE BY SEVERAL NUMBERS
Question:
Find the smallest number exactly divisible by 8, 12, 15 and 20.
Solution:
Required number = LCM of 8, 12, 15 and 20.
8 = 2³.
12 = 2² × 3.
15 = 3 × 5.
20 = 2² × 5.
Take the highest powers:
LCM = 2³ × 3 × 5.
LCM = 8 × 3 × 5.
Answer: 120.
### 20\. LEAST NUMBER LEAVING THE SAME REMAINDER
Question:
Find the least number which, when divided by 12, 15 and 20, leaves a remainder of 5 in each case.
Solution:
First find the LCM:
LCM of 12, 15 and 20 = 60.
Add the common remainder:
60 + 5 = 65.
Answer: 65.
Important Rule:
When a number leaves the same remainder upon division by several divisors, subtract that remainder from the required number and calculate the LCM of the divisors. Add the common remainder to obtain the required number.
### 21\. LEAST NUMBER LEAVING DIFFERENT REMAINDERS
Question:
Find the least number which leaves remainders 2, 3 and 4 when divided by 3, 4 and 5 respectively.
Solution:
Observe:
3 − 2 = 1.
4 − 3 = 1.
5 − 4 = 1.
Therefore, the required number is one less than a common multiple of 3, 4 and 5.
LCM of 3, 4 and 5 = 60.
Required number = 60 − 1.
Answer: 59.
### 22\. IMPORTANT EXAM-BASED CONCEPTS
| Question wording | Required concept |
| ------------------------------------------------- | ------------------- |
| Greatest number that divides | HCF |
| Largest possible equal piece | HCF |
| Maximum number of equal groups | HCF |
| Largest square tile | HCF |
| Smallest number divisible by | LCM |
| Bells ringing together | LCM |
| Events repeating together | LCM |
| Minimum common time | LCM |
| Equal distribution without remainder | HCF |
| Same remainder on division | LCM with adjustment |
| Product of two numbers when HCF and LCM are known | HCF × LCM |
### 23\. COMMON MISTAKES TO AVOID
* Do not confuse factors with multiples.
* Do not select the highest powers when calculating HCF.
* Do not select the lowest powers when calculating LCM.
* Do not apply the product formula directly to three or more numbers.
* In equal distribution problems, carefully identify whether the question asks for the maximum number of groups or the number of objects per group.
* In bell and time problems, convert minutes and seconds into a common unit.
* In decimal problems, correctly restore the decimal places.
* In fraction problems, ensure fractions are in their lowest terms before applying the standard formulas.
### 24\. QUICK REVISION FORMULA SHEET
| Concept | Formula |
| ---------------------- | ---------------------------------------------------- |
| HCF | Product of common prime factors with smallest powers |
| LCM | Product of all prime factors with highest powers |
| Relationship | HCF × LCM = Product of two numbers |
| HCF of Fractions | HCF of Numerators ÷ LCM of Denominators |
| LCM of Fractions | LCM of Numerators ÷ HCF of Denominators |
| Square Tile Side | HCF of Length and Breadth |
| Number of Square Tiles | (Length ÷ Side) × (Breadth ÷ Side) |
| Common Time Interval | LCM of Given Intervals |
### 25\. IMPORTANT MCQs FOR COMPETITIVE EXAMINATIONS
Q1. What is the HCF of 24 and 36?
A) 6 B) 8 C) 12 D) 18
Answer: C) 12.
Q2. Find the LCM of 12 and 15.
A) 30 B) 45 C) 60 D) 90
Answer: C) 60.
Q3. The HCF of two co-prime numbers is:
A) 0 B) 1 C) 2 D) 10
Answer: B) 1.
Q4. The product of two numbers is 720 and their HCF is 6. Find their LCM.
A) 100 B) 110 C) 120 D) 130
Answer: C) 120.
Q5. Find the HCF of 45, 60 and 75.
A) 5 B) 10 C) 15 D) 20
Answer: C) 15.
Q6. Find the LCM of 8, 12 and 16.
A) 24 B) 36 C) 48 D) 64
Answer: C) 48.
Q7. The HCF of 16 and 24 is:
A) 4 B) 6 C) 8 D) 12
Answer: C) 8.
Q8. The LCM of two co-prime numbers 7 and 11 is:
A) 18 B) 77 C) 44 D) 21
Answer: B) 77.
Q9. Find the smallest number divisible by 6, 8 and 12.
A) 12 B) 18 C) 24 D) 48
Answer: C) 24.
Q10. The HCF of 1.2 and 1.8 is:
A) 0.3 B) 0.6 C) 0.9 D) 1.2
Answer: B) 0.6.
Q11. Find the LCM of 18 and 30.
A) 60 B) 75 C) 90 D) 120
Answer: C) 90.
Q12. If the HCF of two numbers is 8 and their product is 960, their LCM is:
A) 100 B) 110 C) 120 D) 140
Answer: C) 120.
Q13. The HCF of 100 and 125 is:
A) 5 B) 10 C) 15 D) 25
Answer: D) 25.
Q14. Find the LCM of 25 and 40.
A) 100 B) 150 C) 200 D) 250
Answer: C) 200.
Q15. Three bells ring at intervals of 10, 15 and 20 seconds. They will ring together again after:
A) 30 seconds B) 40 seconds C) 60 seconds D) 90 seconds
Answer: C) 60 seconds.
Q16. Find the HCF of 72 and 120.
A) 12 B) 18 C) 24 D) 36
Answer: C) 24.
Q17. The LCM of 9, 12 and 15 is:
A) 90 B) 120 C) 180 D) 240
Answer: C) 180.
Q18. Find the HCF of 2/3 and 4/9.
A) 1/3 B) 2/9 C) 4/9 D) 2/3
Answer: B) 2/9.
Q19. Find the LCM of 2/3 and 4/9.
A) 2/3 B) 4/3 C) 4/9 D) 8/9
Answer: B) 4/3.
Q20. The HCF of two consecutive natural numbers is always:
A) 0 B) 1 C) 2 D) 3
Answer: B) 1.
Q21. Find the greatest number that divides 56 and 84 exactly.
A) 7 B) 14 C) 21 D) 28
Answer: D) 28.
Q22. Find the smallest number divisible by 15, 20 and 25.
A) 100 B) 150 C) 300 D) 400
Answer: C) 300.
Q23. If the HCF of two numbers is 12 and their LCM is 180, and one number is 36, the other number is:
A) 48 B) 60 C) 72 D) 90
Answer: B) 60.
Q24. The greatest length of a square tile that can exactly cover a floor of dimensions 120 cm × 180 cm is:
A) 20 cm B) 30 cm C) 60 cm D) 90 cm
Answer: C) 60 cm.
Q25. Find the least number which, when divided by 5, 6 and 8, leaves a remainder of 3 in each case.
A) 120 B) 123 C) 125 D) 128
Answer: B) 123.
### 26\. PRACTICE QUESTIONS WITHOUT ANSWERS
* Find the HCF of 36 and 54.
* Find the LCM of 16 and 24.
* Find the HCF of 72, 96 and 120.
* Find the LCM of 15, 20 and 30.
* Find the smallest number divisible by 12, 18, 24 and 36.
* Find the greatest number that divides 84 and 126 exactly.
* Find the HCF of 1.5 and 2.5.
* Find the LCM of 0.6 and 0.9.
* Three bells ring at intervals of 12, 18 and 24 seconds. When will they ring together again?
* Find the greatest length of a square tile that can cover a floor measuring 240 cm × 360 cm.
* The product of two numbers is 1800 and their HCF is 10. Find their LCM.
* The LCM of two numbers is 240 and their HCF is 8. If one number is 40, find the other number.
* Find the least number which, when divided by 8, 12 and 16, leaves a remainder of 5 in each case.
* Find the HCF of 96, 144 and 192.
* Find the LCM of 21, 28 and 42.
### 27\. ONE-LINER REVISION QUESTIONS
* What is the full form of HCF? – Highest Common Factor.
* What is the full form of LCM? – Least Common Multiple.
* What is another name for HCF? – GCD.
* What is the HCF of two co-prime numbers? – 1.
* What is the LCM of two co-prime numbers? – Their product.
* Which method is also known as the Euclidean Algorithm? – Division Method of HCF.
* Which concept is used in equal distribution? – HCF.
* Which concept is used in repeated events? – LCM.
* What is the HCF of two consecutive natural numbers? – 1.
* What is the relationship between LCM and HCF for two positive integers? – Their product equals the product of the two integers.
### 28\. PREPARATION STRATEGY FOR COMPETITIVE EXAMINATIONS
Candidates preparing for JKSSB, SSC, Banking, Railway, Police, Patwari and other competitive examinations should follow these preparation tips:
* Memorise multiplication tables up to 30.
* Learn prime numbers up to 100.
* Practise prime factorisation regularly.
* Master the division method.
* Memorise the relationship between LCM and HCF.
* Practise word problems involving bells, ropes, tiles and equal distribution.
* Solve at least 20 MCQs daily.
* Practise mental calculations to improve speed.
* Revise important formulas regularly.
* Attempt previous-year questions to understand examination trends.
### 29\. CONCLUSION
LCM and HCF are fundamental topics in Quantitative Aptitude. These concepts are important not only for school-level mathematics but also for various competitive examinations.
Candidates should understand the difference between factors and multiples, practise all methods of calculation and learn how to identify HCF- and LCM-based word problems.
Regular practice of formulas, shortcuts and previous-year questions can help candidates improve their calculation speed, accuracy and overall examination performance.
