HCF and LCM
Imagine you have two ropes, one 12 metres long and the other 18 metres long. You want to cut both ropes into pieces of equal length with no rope left over. What is the longest piece you can cut? The answer is 6 metres, because 6 is the biggest number that divides both 12 and 18 exactly. This biggest number is called the HCF (Highest Common Factor). Now imagine two buses start from the same stop at 8:00 AM. Bus A comes every 12 minutes and Bus B comes every 18 minutes. When will both buses be at the stop together again? In 36 minutes (at 8:36 AM), because 36 is the smallest number that is a multiple of both 12 and 18. This smallest common multiple is called the LCM (Least Common Multiple). HCF and LCM are two of the most useful tools in mathematics. They help you solve sharing problems, scheduling problems, and many other real-life situations. In this chapter, we will learn what HCF and LCM are, how to find them using different methods, and the important relationship between them. This is part of the Playing with Numbers chapter in Grade 6 Maths.
What is HCF and LCM - Grade 6 Maths (Playing with Numbers)?
Common Factors: The common factors of two or more numbers are the factors that all of them share. For example, factors of 12 are {1, 2, 3, 4, 6, 12} and factors of 18 are {1, 2, 3, 6, 9, 18}. The common factors are {1, 2, 3, 6}.
HCF (Highest Common Factor): The HCF of two or more numbers is the largest of their common factors. It is also called the GCD (Greatest Common Divisor). From the example above, the common factors of 12 and 18 are {1, 2, 3, 6}, and the highest one is 6. So HCF(12, 18) = 6.
Common Multiples: The common multiples of two or more numbers are the multiples that all of them share. For example, multiples of 4 are {4, 8, 12, 16, 20, 24, 28, 32, 36, ...} and multiples of 6 are {6, 12, 18, 24, 30, 36, ...}. The common multiples are {12, 24, 36, ...}.
LCM (Least Common Multiple): The LCM of two or more numbers is the smallest of their common multiples. From the example above, the common multiples of 4 and 6 are {12, 24, 36, ...}, and the least (smallest) one is 12. So LCM(4, 6) = 12.
Key difference: HCF is always less than or equal to the given numbers. LCM is always greater than or equal to the given numbers. HCF divides both numbers. Both numbers divide the LCM.
HCF and LCM Formula
Methods to Find HCF:
Method 1: Listing Factors
List all factors of each number. Find the common factors. The greatest one is the HCF.
Method 2: Prime Factorisation
Find the prime factorisation of each number. The HCF is the product of all common prime factors, each taken with the lowest power.
Example: 36 = 2 x 2 x 3 x 3 and 48 = 2 x 2 x 2 x 2 x 3.
Common prime factors: 2 appears at least twice in both, 3 appears at least once in both.
HCF = 2 x 2 x 3 = 12.
Method 3: Long Division Method
Divide the larger number by the smaller. Then divide the smaller number by the remainder. Continue until the remainder is 0. The last divisor is the HCF.
Methods to Find LCM:
Method 1: Listing Multiples
List multiples of each number until you find a common one. The smallest common multiple is the LCM.
Method 2: Prime Factorisation
Find the prime factorisation of each number. The LCM is the product of all prime factors, each taken with the highest power.
Example: 36 = 2 x 2 x 3 x 3 and 48 = 2 x 2 x 2 x 2 x 3.
Take highest power of each prime: 2 appears at most 4 times (in 48), 3 appears at most 2 times (in 36).
LCM = 2 x 2 x 2 x 2 x 3 x 3 = 144.
Method 3: Common Division Method
Write both numbers. Divide by the smallest prime that divides at least one of them. Continue until all quotients are 1. Multiply all the divisors to get the LCM.
HCF-LCM Relationship:
HCF x LCM = Product of the two numbers
For any two numbers a and b: HCF(a,b) x LCM(a,b) = a x b
Verification: HCF(36,48) x LCM(36,48) = 12 x 144 = 1,728. And 36 x 48 = 1,728. Correct!
Derivation and Proof
Let us understand why the prime factorisation method works for both HCF and LCM.
Consider two numbers: 60 and 90.
Prime factorisation: 60 = 2 x 2 x 3 x 5 and 90 = 2 x 3 x 3 x 5.
Think of each number as being built from prime building blocks. 60 has two 2s, one 3, and one 5. 90 has one 2, two 3s, and one 5.
For HCF: We need the largest number that divides BOTH 60 and 90. A number divides 60 only if its prime factors are among {2, 2, 3, 5}. It divides 90 only if its prime factors are among {2, 3, 3, 5}. To divide both, we can only use primes that appear in both factorisations, and we take the minimum count of each.
| Prime | Count in 60 | Count in 90 | Minimum |
| 2 | 2 | 1 | 1 |
| 3 | 1 | 2 | 1 |
| 5 | 1 | 1 | 1 |
HCF = 2 x 3 x 5 = 30.
Check: 60 / 30 = 2 (yes), 90 / 30 = 3 (yes). 30 divides both.
For LCM: We need the smallest number that is a multiple of BOTH 60 and 90. For a number to be a multiple of 60, it must contain at least {2, 2, 3, 5}. For it to be a multiple of 90, it must contain at least {2, 3, 3, 5}. To satisfy both, we take the maximum count of each prime.
| Prime | Count in 60 | Count in 90 | Maximum |
| 2 | 2 | 1 | 2 |
| 3 | 1 | 2 | 2 |
| 5 | 1 | 1 | 1 |
LCM = 2 x 2 x 3 x 3 x 5 = 180.
Check: 180 / 60 = 3 (yes), 180 / 90 = 2 (yes). Both divide 180.
Verify the relationship: HCF x LCM = 30 x 180 = 5,400. Product of numbers = 60 x 90 = 5,400. They match!
This relationship works because when you take the minimum of each prime power (for HCF) and the maximum of each prime power (for LCM), the minimum plus the maximum always equals the sum of the two original powers. So multiplying HCF and LCM reconstructs the product of the original numbers.
Types and Properties
Here are the types of problems you will encounter:
Type 1: Finding HCF by Listing Factors - List all factors of each number, find common factors, identify the highest. Best for small numbers.
Type 2: Finding HCF by Prime Factorisation - Find prime factorisation of each number, take common primes with lowest powers. Best for larger numbers.
Type 3: Finding HCF by Long Division - Repeatedly divide the larger by smaller, then divisor by remainder, until remainder is 0. The last divisor is HCF. Most efficient for very large numbers.
Type 4: Finding LCM by Listing Multiples - List multiples of each number until you find a common one. Best for small numbers.
Type 5: Finding LCM by Prime Factorisation - Find prime factorisation, take all primes with highest powers. Best for larger numbers.
Type 6: Finding LCM by Common Division - Divide by primes repeatedly. Multiply all divisors for LCM. Good for finding LCM of three or more numbers.
Type 7: Using the HCF-LCM Relationship - If you know one of HCF or LCM and the two numbers, find the other using: HCF x LCM = Product of the two numbers.
Type 8: Word Problems - Real-life problems involving cutting, sharing, scheduling, arranging, or finding when events coincide. These require identifying whether to use HCF or LCM.
Solved Examples
Example 1: Example 1: HCF by Listing Factors
Problem: Find the HCF of 24 and 36.
Solution:
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 (the highest common factor)
Example 2: Example 2: HCF by Prime Factorisation
Problem: Find the HCF of 72 and 120.
Solution:
72 = 2 x 2 x 2 x 3 x 3
120 = 2 x 2 x 2 x 3 x 5
Common prime factors with minimum powers:
2 appears 3 times in both, so take 2 x 2 x 2
3 appears once in 120 and twice in 72, so take 3 (minimum is 1 time)
5 appears only in 120, not common, so skip
HCF = 2 x 2 x 2 x 3 = 24
Example 3: Example 3: HCF by Long Division
Problem: Find the HCF of 56 and 98 using the long division method.
Solution:
Step 1: Divide 98 by 56. 98 = 56 x 1 + 42 (remainder 42)
Step 2: Divide 56 by 42. 56 = 42 x 1 + 14 (remainder 14)
Step 3: Divide 42 by 14. 42 = 14 x 3 + 0 (remainder 0)
The remainder is 0, so the last divisor (14) is the HCF.
HCF(56, 98) = 14
Verification: 56 / 14 = 4 and 98 / 14 = 7. Both divide exactly.
Example 4: Example 4: LCM by Listing Multiples
Problem: Find the LCM of 6 and 8.
Solution:
Multiples of 6: 6, 12, 18, 24, 30, 36, ...
Multiples of 8: 8, 16, 24, 32, 40, ...
The first common multiple is 24.
LCM(6, 8) = 24
Example 5: Example 5: LCM by Prime Factorisation
Problem: Find the LCM of 36 and 48.
Solution:
36 = 2 x 2 x 3 x 3
48 = 2 x 2 x 2 x 2 x 3
Take each prime with its highest power:
2 appears at most 4 times (in 48), so take 2 x 2 x 2 x 2
3 appears at most 2 times (in 36), so take 3 x 3
LCM = 2 x 2 x 2 x 2 x 3 x 3 = 16 x 9 = 144
Example 6: Example 6: LCM by Common Division
Problem: Find the LCM of 12, 15, and 20.
Solution:
Write all numbers. Divide by the smallest prime that divides at least one:
2 | 12, 15, 20
2 | 6, 15, 10
3 | 3, 15, 5
5 | 1, 5, 5
| 1, 1, 1
LCM = 2 x 2 x 3 x 5 = 60
Verification: 60/12=5, 60/15=4, 60/20=3. All divide exactly.
Example 7: Example 7: Using the HCF-LCM Relationship
Problem: The HCF of two numbers is 6 and their LCM is 180. If one number is 36, find the other number.
Solution:
Using the relationship: HCF x LCM = Product of the two numbers
6 x 180 = 36 x other number
1,080 = 36 x other number
Other number = 1,080 / 36 = 30
Verification: HCF(36, 30) = 6, LCM(36, 30) = 180. And 6 x 180 = 1,080 = 36 x 30. Correct!
Example 8: Example 8: Word Problem - Cutting Ribbons (HCF)
Problem: Two ribbons are 16 cm and 24 cm long. They need to be cut into pieces of equal length without any ribbon left over. What is the greatest possible length of each piece?
Solution:
We need the greatest length that divides both 16 and 24 exactly. This is the HCF.
Factors of 16: 1, 2, 4, 8, 16
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Common factors: 1, 2, 4, 8
HCF = 8
The greatest possible length of each piece is 8 cm.
From the 16 cm ribbon: 16/8 = 2 pieces. From the 24 cm ribbon: 24/8 = 3 pieces. Total = 5 pieces.
Example 9: Example 9: Word Problem - Bells Ringing (LCM)
Problem: Two bells ring at intervals of 8 minutes and 12 minutes respectively. If they ring together at 9:00 AM, when will they ring together again?
Solution:
They ring together again after a time that is a common multiple of 8 and 12. We need the smallest such time, so we find the LCM.
8 = 2 x 2 x 2
12 = 2 x 2 x 3
LCM = 2 x 2 x 2 x 3 = 24 minutes
They will ring together again at 9:24 AM.
Example 10: Example 10: Word Problem - Arranging Students (HCF)
Problem: A school has 120 boys and 150 girls. They want to arrange them in rows such that each row has the same number of students and each row has only boys or only girls. What is the maximum number of students in each row?
Solution:
Each row must have the same number of students, and this number must divide both 120 and 150 exactly. We need the greatest such number, so we find HCF(120, 150).
120 = 2 x 2 x 2 x 3 x 5
150 = 2 x 3 x 5 x 5
Common factors with minimum powers: 2 (once), 3 (once), 5 (once)
HCF = 2 x 3 x 5 = 30
Each row will have 30 students.
Boys rows: 120 / 30 = 4 rows. Girls rows: 150 / 30 = 5 rows. Total rows = 9.
Real-World Applications
HCF and LCM are used in many real-life situations. Whenever you need to divide or share things equally and want the biggest possible groups, you use HCF. Whenever you need to find when two or more events will happen at the same time, you use LCM.
HCF applications: Cutting ropes, ribbons, or pipes into equal pieces. Dividing groups of items equally. Simplifying fractions (divide numerator and denominator by their HCF). Tiling a rectangular floor with square tiles of the largest possible size.
LCM applications: Finding when two events with different cycles will coincide (like two traffic lights, two buses, or two bells). Finding the smallest container that can be completely filled using either of two different-sized cups. Finding common denominators when adding fractions. Scheduling recurring events.
In cooking, if one recipe uses 3 eggs and another uses 4 eggs, and you want to make both recipes using the same number of eggs, you need LCM(3, 4) = 12 eggs. So you make 4 batches of the first recipe and 3 batches of the second.
In sports scheduling, if Team A plays every 3 days and Team B plays every 5 days, they will play on the same day every LCM(3, 5) = 15 days.
In fractions, to add 1/6 and 1/8, you need a common denominator. The LCM of 6 and 8 is 24. So you convert to 4/24 + 3/24 = 7/24. LCM makes fraction operations possible.
Key Points to Remember
- HCF (Highest Common Factor) is the greatest number that divides two or more numbers exactly.
- LCM (Least Common Multiple) is the smallest number that is a multiple of two or more numbers.
- Three methods to find HCF: listing factors, prime factorisation, long division.
- Three methods to find LCM: listing multiples, prime factorisation, common division.
- For HCF by prime factorisation: take common primes with the lowest power.
- For LCM by prime factorisation: take all primes with the highest power.
- HCF x LCM = Product of the two numbers (for two numbers only).
- HCF of two numbers is always less than or equal to both numbers.
- LCM of two numbers is always greater than or equal to both numbers.
- If two numbers are co-prime (HCF = 1), then LCM = product of the two numbers.
- Use HCF when dividing into equal groups. Use LCM when finding common timing or common denominators.
Practice Problems
- Find the HCF of 42 and 70 using the prime factorisation method.
- Find the LCM of 15 and 25 using prime factorisation.
- Find the HCF of 84 and 126 using the long division method.
- Find the LCM of 8, 12, and 18 using the common division method.
- The HCF of two numbers is 8 and their LCM is 120. If one number is 24, find the other.
- Three ropes are 36 cm, 48 cm, and 60 cm long. What is the greatest length of equal pieces they can be cut into?
- Three friends visit a park every 4, 6, and 9 days respectively. If they all visit today, after how many days will they all visit on the same day again?
- Find two numbers whose HCF is 5 and LCM is 60.
Frequently Asked Questions
Q1. How do I know whether to use HCF or LCM in a word problem?
Use HCF when the problem involves dividing or splitting things into equal groups and finding the biggest possible group size. Key words: 'greatest', 'largest piece', 'maximum', 'divide equally'. Use LCM when the problem involves finding when events coincide or finding the smallest common quantity. Key words: 'together again', 'at the same time', 'least', 'minimum', 'common denominator'.
Q2. Can HCF be greater than LCM?
No, never. The HCF is always less than or equal to the LCM. They are equal only when both numbers are the same. For example, HCF(12, 12) = 12 and LCM(12, 12) = 12. In all other cases, HCF < LCM.
Q3. Does the HCF x LCM formula work for three numbers?
No, the formula HCF x LCM = Product of the numbers works only for TWO numbers. For three or more numbers, this formula does not hold. You need to find HCF and LCM of three numbers using their individual methods (prime factorisation or common division).
Q4. What is the HCF of two co-prime numbers?
The HCF of two co-prime numbers is always 1. By definition, co-prime numbers have no common factor other than 1. For example, HCF(8, 15) = 1 because 8 and 15 are co-prime. In this case, LCM(8, 15) = 8 x 15 = 120.
Q5. What is the LCM of two consecutive numbers?
The LCM of two consecutive numbers is always their product. For example, LCM(4, 5) = 20 = 4 x 5, and LCM(7, 8) = 56 = 7 x 8. This is because consecutive numbers are always co-prime (their HCF is 1), so LCM = product.
Q6. What is the HCF of a number and its multiple?
If B is a multiple of A, then HCF(A, B) = A. For example, HCF(6, 24) = 6 because 24 is a multiple of 6. Similarly, LCM(A, B) = B when B is a multiple of A.
Q7. Which method for finding HCF is the fastest?
For small numbers (less than 100), listing factors is quick. For medium numbers, prime factorisation works well. For large numbers, the long division method is the fastest because it does not require finding prime factors. Each method gives the same answer; choose based on the size of the numbers.
Q8. Can HCF or LCM be 0?
HCF is defined for positive numbers, and it is always at least 1 (since 1 is a factor of every number). LCM is also always at least as large as the bigger of the two numbers. Neither HCF nor LCM can be 0 for positive whole numbers.
Q9. How are HCF and LCM used in fractions?
HCF is used to simplify fractions. To simplify 12/18, find HCF(12, 18) = 6 and divide both by 6 to get 2/3. LCM is used to find common denominators for adding or subtracting fractions. To add 1/4 + 1/6, find LCM(4, 6) = 12, then convert: 3/12 + 2/12 = 5/12.
Q10. If one number is a factor of the other, what are their HCF and LCM?
If A is a factor of B (say B = k x A for some whole number k), then HCF(A, B) = A and LCM(A, B) = B. For example, for 5 and 20: HCF = 5, LCM = 20. Check: 5 x 20 = 100 = 5 x 20. The relationship holds.
