HCF and LCM are two important topics in mathematics. They are used in number system, fractions, simplification, time and work, bells and clocks, arrangements, grouping, and many competitive exam questions. If you understand the basic meaning of factor, multiple, HCF, and LCM, most questions from this topic become easy.
HCF means Highest Common Factor. It is the greatest number that can divide two or more numbers exactly. LCM means Least Common Multiple. It is the smallest number that is exactly divisible by two or more given numbers. In many books, HCF is also called GCD, which means Greatest Common Divisor. LCM is also called Least Common Multiple or Lowest Common Multiple.
This lesson explains HCF and LCM step by step with formulas, methods, solved examples, shortcut ideas, fraction and decimal cases, and common exam-style questions.
Basic Terms Used in HCF and LCM
Factor
A factor of a number is a number that divides it completely without leaving any remainder. For example, 3 is a factor of 12 because 12 divided by 3 gives 4 with no remainder. The factors of 12 are 1, 2, 3, 4, 6, and 12.
Multiple
A multiple of a number is obtained by multiplying that number by 1, 2, 3, 4, and so on. For example, the multiples of 5 are 5, 10, 15, 20, 25, 30, and so on.
Common Factor
A common factor is a factor that is present in two or more numbers. For example, factors of 18 are 1, 2, 3, 6, 9, 18. Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The common factors are 1, 2, 3, and 6.
Common Multiple
A common multiple is a number that is a multiple of two or more numbers. For example, multiples of 4 are 4, 8, 12, 16, 20, 24. Multiples of 6 are 6, 12, 18, 24, 30. The common multiples include 12 and 24.
What Is HCF?
HCF of two or more numbers is the greatest number that divides all the given numbers exactly. It is useful when we need to divide things into equal groups of maximum possible size.
Example: Find the HCF of 12 and 18.
Factors of 12 are 1, 2, 3, 4, 6, 12.
Factors of 18 are 1, 2, 3, 6, 9, 18.
Common factors are 1, 2, 3, and 6.
The greatest common factor is 6.
So, HCF of 12 and 18 is 6.
What Is LCM?
LCM of two or more numbers is the smallest number that is exactly divisible by each given number. It is useful when repeated events have to meet at the same time.
Example: Find the LCM of 4 and 6.
Multiples of 4 are 4, 8, 12, 16, 20, 24.
Multiples of 6 are 6, 12, 18, 24, 30.
The smallest common multiple is 12.
So, LCM of 4 and 6 is 12.
Difference Between HCF and LCM
| Point | HCF | LCM |
|---|---|---|
| Full form | Highest Common Factor | Least Common Multiple |
| Meaning | Greatest number that divides the given numbers | Smallest number divisible by the given numbers |
| Based on | Factors | Multiples |
| Use | Maximum equal grouping | Minimum common repetition |
| Example | HCF of 12 and 18 is 6 | LCM of 12 and 18 is 36 |
Methods to Find HCF
1. Listing Factors Method
This method is best for small numbers. Write the factors of each number, find the common factors, and select the greatest one.
Example: Find the HCF of 16 and 24.
Factors of 16 are 1, 2, 4, 8, 16.
Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.
Common factors are 1, 2, 4, and 8.
HCF is 8.
2. Prime Factorization Method
In this method, write each number as a product of prime factors. Then multiply the common prime factors with the lowest powers.
Example: Find the HCF of 72, 108, and 210.
72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²
108 = 2 × 2 × 3 × 3 × 3 = 2² × 3³
210 = 2 × 3 × 5 × 7
The common prime factors are 2 and 3.
The lowest power of 2 is 2¹ and the lowest power of 3 is 3¹.
HCF = 2 × 3 = 6.
3. Division Method or Euclidean Method
This method is useful for large numbers. Divide the larger number by the smaller number. Then divide the previous divisor by the remainder. Continue until the remainder becomes zero. The last divisor is the HCF.
Example: Find the HCF of 148 and 185.
185 ÷ 148 gives remainder 37.
148 ÷ 37 gives remainder 0.
So, HCF of 148 and 185 is 37.
Example: Find the HCF of 513, 1134, and 1215.
First find HCF of 1134 and 1215.
1215 ÷ 1134 gives remainder 81.
1134 ÷ 81 gives remainder 0.
So, HCF of 1134 and 1215 is 81.
Now find HCF of 513 and 81.
513 ÷ 81 gives remainder 27.
81 ÷ 27 gives remainder 0.
So, HCF of 513, 1134, and 1215 is 27.
Methods to Find LCM
1. Listing Multiples Method
This method is useful for small numbers. Write multiples of each number and choose the smallest common multiple.
Example: Find the LCM of 8 and 12.
Multiples of 8 are 8, 16, 24, 32, 40.
Multiples of 12 are 12, 24, 36, 48.
The smallest common multiple is 24.
So, LCM of 8 and 12 is 24.
2. Prime Factorization Method
In this method, write all numbers as prime factors. For LCM, take all prime factors with the highest powers and multiply them.
Example: Find the LCM of 72, 108, and 2100.
72 = 2³ × 3²
108 = 2² × 3³
2100 = 2² × 3 × 5² × 7
For LCM, take the highest powers: 2³, 3³, 5², and 7.
LCM = 2³ × 3³ × 5² × 7
LCM = 8 × 27 × 25 × 7 = 37800.
3. Common Division Method
In this method, write all given numbers in a row. Divide by a prime number that divides at least one of the numbers. Keep dividing until all numbers become 1. The product of all divisors is the LCM.
Example: Find the LCM of 16, 24, 36, and 54.
16 = 2⁴
24 = 2³ × 3
36 = 2² × 3²
54 = 2 × 3³
LCM = 2⁴ × 3³ = 16 × 27 = 432.
Important Formula of HCF and LCM
For two numbers:
Product of two numbers = HCF × LCM
If the two numbers are A and B, then:
A × B = HCF(A, B) × LCM(A, B)
Example: Numbers are 12 and 15.
HCF of 12 and 15 is 3.
LCM of 12 and 15 is 60.
Product of numbers = 12 × 15 = 180.
Product of HCF and LCM = 3 × 60 = 180.
This formula is very useful in exam questions where the HCF, LCM, or product of two numbers is given.
Co-Prime Numbers
Two numbers are called co-prime numbers if their HCF is 1. Co-prime numbers do not need to be prime numbers themselves.
Example: 8 and 15 are co-prime because HCF of 8 and 15 is 1.
Here, 8 is not prime and 15 is not prime, but they are co-prime with each other.
If two numbers are co-prime, their LCM is equal to their product.
Example: LCM of 7 and 9 = 7 × 9 = 63 because HCF of 7 and 9 is 1.
HCF and LCM of Fractions
Before finding HCF or LCM of fractions, first convert the fractions into their simplest form.
HCF of fractions = HCF of numerators ÷ LCM of denominators
LCM of fractions = LCM of numerators ÷ HCF of denominators
Example: Find the HCF of 2/3, 10/27, 8/9, and 16/81.
Numerators are 2, 10, 8, and 16.
Denominators are 3, 27, 9, and 81.
HCF of numerators = 2.
LCM of denominators = 81.
HCF of fractions = 2/81.
Example: Find the LCM of 1/2, 5/6, and 5/4.
LCM of numerators 1, 5, 5 is 5.
HCF of denominators 2, 6, 4 is 2.
LCM of fractions = 5/2.
HCF and LCM of Decimal Numbers
To find HCF and LCM of decimals, first make the same number of decimal places by adding zeros if required. Then remove the decimal point and find HCF or LCM like whole numbers. Finally, put the decimal point back according to the number of decimal places.
Example: Find HCF and LCM of 0.63, 1.05, and 2.10.
All numbers have two decimal places.
Remove decimals: 63, 105, 210.
HCF of 63, 105, and 210 is 21.
So, HCF of 0.63, 1.05, and 2.10 is 0.21.
LCM of 63, 105, and 210 is 630.
So, LCM of 0.63, 1.05, and 2.10 is 6.30.
How to Decide Whether to Use HCF or LCM
Many students know the formulas but get confused in word problems. The easiest way is to understand the meaning of the question.
Use HCF when the question asks for the greatest number, largest size, maximum length, equal groups, equal packets, or maximum capacity.
Use LCM when the question asks for the least number, smallest number, first common time, repeated events meeting together, bells ringing together, or minimum quantity divisible by all given numbers.
| Question clue | Use |
|---|---|
| Greatest number that divides | HCF |
| Largest possible equal group | HCF |
| Maximum length of tape or rope | HCF |
| Least number divisible by given numbers | LCM |
| Bells ringing together again | LCM |
| Minimum number of items arranged in rows | LCM |
Exam Type 1: Find the Greatest Number That Divides Given Numbers
Question: Find the greatest number that divides 24, 36, and 60 exactly.
Solution: We need the greatest number that divides all given numbers, so this is an HCF question.
24 = 2³ × 3
36 = 2² × 3²
60 = 2² × 3 × 5
Common factors with lowest powers are 2² and 3.
HCF = 4 × 3 = 12.
Exam Type 2: Find the Least Number Divisible by Given Numbers
Question: Find the least number divisible by 12, 15, 20, and 30.
Solution: We need the least number divisible by all numbers, so this is an LCM question.
12 = 2² × 3
15 = 3 × 5
20 = 2² × 5
30 = 2 × 3 × 5
LCM = 2² × 3 × 5 = 60.
Exam Type 3: Same Remainder in Each Case
Question: Find the largest number which divides 62, 132, and 237 leaving the same remainder in each case.
Solution: When the same remainder is left, take the differences of the numbers and find their HCF.
132 – 62 = 70
237 – 132 = 105
237 – 62 = 175
HCF of 70, 105, and 175 is 35.
So, the required number is 35.
Exam Type 4: Different Remainders
Question: Find the greatest number which divides 259 and 465 leaving remainders 4 and 6 respectively.
Solution: Subtract the remainders from the numbers.
259 – 4 = 255
465 – 6 = 459
Now find HCF of 255 and 459.
459 ÷ 255 gives remainder 204.
255 ÷ 204 gives remainder 51.
204 ÷ 51 gives remainder 0.
So, HCF is 51.
The required number is 51.
Exam Type 5: Same Remainder After Division
Question: Find the least number which when divided by 6, 14, 18, and 22 leaves remainder 4 in each case.
Solution: First find LCM of 6, 14, 18, and 22.
6 = 2 × 3
14 = 2 × 7
18 = 2 × 3²
22 = 2 × 11
LCM = 2 × 3² × 7 × 11 = 1386.
Since the required number leaves remainder 4 in each case, add 4.
Required number = 1386 + 4 = 1390.
Exam Type 6: Ratio and HCF Given
Question: The HCF of two numbers is 13 and the numbers are in the ratio 15:11. Find the numbers.
Solution: Let the numbers be 15x and 11x.
Since 15 and 11 are co-prime, x is the HCF.
So, x = 13.
First number = 15 × 13 = 195.
Second number = 11 × 13 = 143.
The numbers are 195 and 143.
Exam Type 7: Product and HCF Given
Question: The product of two numbers is 4107 and their HCF is 37. Find the LCM.
Solution: Product of two numbers = HCF × LCM.
4107 = 37 × LCM
LCM = 4107 ÷ 37 = 111.
So, the LCM is 111.
Exam Type 8: Bells Ringing Together
Question: Three bells ring at intervals of 12 seconds, 18 seconds, and 30 seconds. If they ring together now, after how many seconds will they ring together again?
Solution: Such repeated-time questions are solved by LCM.
12 = 2² × 3
18 = 2 × 3²
30 = 2 × 3 × 5
LCM = 2² × 3² × 5 = 180.
So, the bells will ring together again after 180 seconds.
Exam Type 9: Arrangement in Rows
Question: What is the least number of students that can be arranged in rows of 12, 15, or 20 students exactly?
Solution: The number must be divisible by 12, 15, and 20. So we find LCM.
LCM of 12, 15, and 20 is 60.
So, the least number of students is 60.
Exam Type 10: Maximum Length Problem
Question: Three ropes are 24 m, 36 m, and 60 m long. What is the greatest length of each piece if all ropes are cut into equal pieces without waste?
Solution: We need the greatest equal length, so use HCF.
HCF of 24, 36, and 60 is 12.
So, each piece should be 12 m long.
HCF and LCM Short Tricks
- If two numbers are co-prime, HCF is 1 and LCM is their product.
- HCF can never be greater than the smallest given number.
- LCM can never be smaller than the largest given number.
- HCF of two prime numbers is usually 1, unless both numbers are the same.
- For two numbers, always remember: product = HCF × LCM.
- For remainder questions, subtract the remainder first and then find HCF.
- For repeated events, bells, clocks, and meeting again questions, use LCM.
Common Mistakes in HCF and LCM
One common mistake is using HCF where LCM is required. If the question asks for the least number divisible by many numbers, it is LCM, not HCF.
Another mistake is forgetting to simplify fractions before applying HCF and LCM formulas for fractions. Always reduce fractions to their lowest terms first.
Students also make mistakes in decimal questions by using different decimal places. Make all decimal places equal before removing the decimal point.
In prime factorization, remember that HCF uses the lowest powers of common factors, while LCM uses the highest powers of all factors.
Practice Questions
- Find the HCF of 48, 72, and 96.
- Find the LCM of 18, 24, and 30.
- Find the greatest number that divides 43, 91, and 183 leaving the same remainder.
- Find the least number divisible by 8, 12, 15, and 20.
- The HCF of two numbers is 18 and the numbers are in the ratio 5:7. Find the numbers.
- Three bells ring at intervals of 10, 15, and 25 seconds. When will they ring together again?
- Find the HCF of 0.24, 0.36, and 0.60.
- Find the LCM of 2/3, 4/5, and 8/15.
Answers to Practice Questions
- HCF of 48, 72, and 96 is 24.
- LCM of 18, 24, and 30 is 360.
- Differences are 48, 92, and 140. HCF is 4.
- LCM of 8, 12, 15, and 20 is 120.
- Numbers are 90 and 126.
- LCM of 10, 15, and 25 is 150 seconds.
- HCF of 0.24, 0.36, and 0.60 is 0.12.
- LCM of fractions = LCM of numerators 2, 4, 8 divided by HCF of denominators 3, 5, 15 = 8/1 = 8.
Frequently Asked Questions
What is the full form of HCF?
HCF stands for Highest Common Factor. It is the greatest number that divides two or more numbers exactly.
What is the full form of LCM?
LCM stands for Least Common Multiple. It is the smallest number that is divisible by two or more given numbers.
Is HCF the same as GCD?
Yes. HCF and GCD mean the same thing. HCF means Highest Common Factor and GCD means Greatest Common Divisor.
Can HCF be greater than LCM?
No. For positive whole numbers, HCF cannot be greater than LCM. HCF is usually smaller and LCM is usually larger.
What is the HCF of two co-prime numbers?
The HCF of two co-prime numbers is 1.
What is the LCM of two co-prime numbers?
The LCM of two co-prime numbers is equal to their product.
Where are HCF and LCM used in real life?
HCF is used for making equal groups, cutting things into maximum equal lengths, and distributing items equally. LCM is used for repeated events, schedules, bells ringing together, and finding common denominators in fractions.
How can I quickly identify HCF or LCM in a word problem?
If the question asks for the greatest number, maximum size, or largest equal group, use HCF. If it asks for the least number, first common time, or smallest number divisible by given numbers, use LCM.
More HCF and LCM Exam Tips
In exam questions, HCF and LCM are often hidden inside word problems. If the question asks for the greatest number that can divide many numbers, choose HCF. If it asks for the least number that is divisible by many numbers, choose LCM. This one decision is the most important part of the topic.
For remainder based questions, subtract the given remainder from each number first. Then find the HCF of the new numbers. For repeated event questions, such as bells ringing together or lights blinking together, find the LCM of the given time intervals.
When two numbers are given with their HCF and LCM, remember that product of the two numbers is equal to HCF multiplied by LCM. If the ratio of two numbers is also given, write the numbers as ax and bx and solve using the HCF or product condition.
Example
The HCF of two numbers is 12 and the numbers are in the ratio 5:7. Find the numbers.
Let the numbers be 5x and 7x. Since 5 and 7 are co-prime, x is the HCF. So x = 12. The numbers are 60 and 84.
This method is very useful in SSC, banking and railway aptitude questions because it avoids unnecessary trial and error.
Conclusion
HCF and LCM are simple once the meaning is clear. HCF is about the greatest common divisor, while LCM is about the smallest common multiple.
Learn the methods, practice different question types, and always read the wording carefully before deciding whether the question needs HCF or LCM.