HCF / GCD and LCM

The full form of HCF/GCD is the Highest Common Factor/Greatest Common Divisor, while the full form of LCM is the Least Common Multiple.

HCF/GCD: The largest number that divides two or more numbers exactly without leaving any remainder.
LCM: The smallest number that is exactly divisible by two or more numbers.

Example: Find the HCF & LCM of 6 and 18.

To find the HCF
Divisors of 6 = 1, 2, 3, 6
Divisors of 18 = 1, 2, 3, 6, 9, 18
HCF = greatest common divisor
HCF = 6

To find the LCM
Multiple of 6 = 6, 12, 18, 24, 30, …
Multiple of 18 = 18, 36, 54, …
LCM = first common multiple (least common multiple)
LCM = 18

HCF / GCD and LCM Formula

To find the HCF and LCM formulas, let's assume that the numbers given are a and b. The relationship between HCF and LCM states that the product of a and b is equal to the product of HCF and LCM.

(LCM of two numbers) × (HCF of two numbers) = Product of two numbers

Mathematically, this can be written as:

LCM(a, b) × HCF(a, b) = a × b
Note: This formula is valid only for two numbers. It does not always hold true for more than two numbers.

How to Find HCF and LCM?

These are the most famous methods used to calculate HCF and LCM :

1. HCF by Division Method

The easiest way to understand how to find HCF by the Division Method is by going back to simple division.

The following are the steps for better understanding this method :

Step 1: Take the smaller number as the divisor and the larger number as a dividend.
Step 2: Perform division. If you get the remainder as 0, then the divisor is the HCF of the given numbers.
Step 3: If you get a remainder other than 0 then take the remainder as the new divisor and the previous divisor as the new dividend.
Step 4: Perform steps 2 and step 3 until you get the remainder as 0.

Example: Find out the HCF of 36 and 48.

Solution:

Using the division method for HCF
Hence, HCF = 12

2. LCM by Division Method

The following steps can be followed to find the Least Common Multiple by the Division Method:

Step 1: Check whether the given numbers are divisible by 2 or not.
Step 2: If the number is divisible by 2 then divide and again check for the same. If the numbers are not divisible by 2 then check 3, and so on.
Step 3: Perform step 2 until you get 1 in the end.

Example: Find out the LCM of 36 and 48.

Solution:

Using the division method for LCM
Hence, LCM = 2 × 2 × 2 × 2 × 3 × 3 = 144

3. HCF by Prime Factorization

Finding HCF by Prime Factorization can be done by following the given steps:

Step 1: Find out the prime factors of the given number.
Step 2: Check the occurrence of a particular factor. Find out the common factors and choose them in HCF.
Step 3: Multiply the occurrence of common factors. And this will be the HCF Of the given numbers.

Example: Find out the HCF of 18 and 90.

Solution:

Prime factors of 18 = 2 × 3 × 3
Prime factors of 90 = 2 × 3 × 3 × 5
Now, HCF = 2 × 3 × 3 = 18

4. LCM by Prime Factorization

Finding LCM by Prime Factorization is done by following the given steps:

Step 1: Find out the prime factors of the given number.
Step 2: Check the occurrence of a particular factor. If a particular factor has occurred multiple times in the given number, then choose the maximum occurrence of the factor in LCM. It can also be found out by checking the powers of the factors. The factor having greater power will be chosen between the numbers.
Step 3: Multiply all the maximum occurrences of a particular factor. And this will be the LCM Of the given numbers.

Example: Find out the LCM of 18 and 90.

Solution:

Prime factors of 18 = 2 × 3 × 3
Prime factors of 90 = 2 × 3 × 3 × 5
Now, LCM = 2 × 3 × 3 × 5 = 90
Alternate method:
Prime factors of 18 = 2 × 3 × 3
Prime factors of 18 = 2¹ × 3²
Prime factors of 90 = 2 × 3 × 3 × 5
Prime factors of 90 = 2¹ × 3²× 5¹
Chosen factors for LCM = 2¹ × 3² × 5¹
Therefore, LCM = 2 × 9 × 5 = 90.

Rule to Find the HCF of Fractions

To find the HCF of two or more fractions, first find the HCF of the numerators and the LCM of the denominators, then divide them.

HCF of fractions = HCF of the numerators ÷ LCM of the denominators

Example: Find the HCF of 3/4 and 9/10.

Solution:

HCF of numerators = HCF of 3 and 9 = 3
LCM of denominators = LCM of 4 and 10 = 20
HCF = 3/20

Rule to Find the LCM of Fractions

To find the LCM of two or more fractions, first find the LCM of the numerators and the HCF of the denominators, then divide them.

LCM of fractions = LCM of the numerators ÷ HCF of the denominators

Example: Find the LCM of 3/4 and 9/10.

Solution:

LCM of numerators = LCM of 3 and 9 = 9
HCF of denominators = HCF of 4 and 10 = 2
LCM = 9/2

How to Find HCF
How to Find GCD of more than 2 Number?
Finding GCD by Prime Factorization
Properties of GCD

Questions and Answers

Question 1: Find out the LCM and HCF of 18, 30, and 90 by prime factorization.

Solution:

Prime factors of 18 = 2 × 3 × 3
Prime factors of 30 = 2 × 3 × 5
Prime factors of 90 = 2 × 3 × 3 × 5
LCM: 2 × 3 × 3 × 5 = 90
HCF: 2 × 3 = 6

Question 2: Find out the LCM and HCF of 318 and 504.

Solution:

Prime factors of 318 = 2 × 3 × 53
Prime factors of 504 = 2 × 2 × 2 × 3 × 3 × 7
LCM: 2 × 2 × 2 × 3 × 3 × 7 × 53 = 26712
HCF: 2 × 3 = 6

Question 3: Find out the HCF of 24 and 36.
Solution:

Let's find out the HCF of 24 and 36 by division method,
Therefore, HCF = 2 × 2 × 3 = 12

Question 4: Find out the LCM of 24 and 36.

Solution:

Let's find out the LCM of 24 and 36 by division method,
Therefore, LCM = 2 × 2 × 3 × 2 × 3 = 72

Question 5: Find out the LCM and HCF of 15 and 70. Also, verify the relationship between LCM, HCF, and the given numbers.

Solution:

Prime factors of 15 = 3 × 5
Prime factors of 70 = 2 × 5 × 7
LCM: 2 × 3 × 5 × 7
HCF: 5
Verifying the relationship:
LCM × HCF = 2 × 3 × 5 × 5 × 7 = 1050
Product of two numbers = 15 × 70 = 1050
From above you can see that,
LCM (15, 70) × HCF(15, 70) = Product of two numbers
Hence Verified.

Practice Questions

Question 1: Find the HCF of 36 and 60.

Question 2: What is the LCM of 12, 18, and 24?

Question 3: Two numbers have an HCF of 8 and an LCM of 96. If one of the numbers is 32, find the other number.

Question 4: Calculate the HCF and LCM of 45 and 75.

Question 5: The product of two numbers is 2400, and their HCF is 20. Find their LCM.

Question 6: Find the HCF of 72, 108, and 144.

Question 7: Two cyclists are riding on circular tracks. One completes a round in 12 minutes, and the other in 18 minutes. After how many minutes will both cyclists meet at the starting point if they start together? (Hint: Find the LCM of their times.)

Question 8: Three friends have ropes of lengths 24 meters, 36 meters, and 48 meters. They want to cut their ropes into equally smaller pieces without any leftovers. What is the maximum possible length of each smaller piece they can cut? (Hint: Find the HCF of the rope lengths.)

Answer Key
Ans 1: HCF = 12
Ans 2: LCM = 72
Ans 3: Other number = 24
Ans 4: HCF = 15, LCM = 225
Ans 5: LCM = 120
Ans 6: HCF = 36
Ans 7: LCM = 36 minutes
Ans 8: HCF = 12 meters

HCF / GCD and LCM Formula

How to Find HCF and LCM?

1. HCF by Division Method

2. LCM by Division Method

3. HCF by Prime Factorization

4. LCM by Prime Factorization

Rule to Find the HCF of Fractions

Rule to Find the LCM of Fractions

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Questions and Answers

Practice Questions

Answer Key

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