HCF and LCM

The full forms of H.C.F. and L.C.M. are, Highest Common factor and Least Common Multiple, respectively. The H.C.F. defines the greatest factor present in between given two or more numbers, whereas L.C.M. defines the least number which is exactly divisible by two or more numbers. H.C.F. is also called the greatest common factor (GCF) and LCM is also called the Least Common Divisor.

To find H.C.F. and L.C.M., we have two important methods which are the Prime factorisation method and division method. We have learned both methods in our earlier classes. The shortcut method to find both H.C.F. and L.C.M. is a division method. Let us learn the relationship between HCF and LCM with the help of the formula here. Also, we will solve some problems based on these two concepts to understand better. The article here is very helpful for primary and secondary classes students such as Class 4, Class 5, Class 6, Class 7 and Class 8.

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HCF and LCM Definition

We know that the factors of a number are exact divisors of that particular number. Let’s proceed to the highest common factor (H.C.F.) and the least common multiple (L.C.M.).

HCF (Highest Common Factor)
As the rules of mathematics dictate, the greatest common divisor or the gcd of two or more positive integers happens to be the largest positive integer that divides the numbers without leaving a remainder. For example, take 8 and 12. The H.C.F. of 8 and 12 will be 4 because the highest number that can divide both 8 and 12 is 4.

LCM (Least Common Multiple)
In arithmetic, the least common multiple or LCM of two numbers say a and b, is denoted as LCM (a,b). And the LCM is the smallest or least positive integer that is divisible by both a and b. For example, let us take two positive integers 4 and 6.

Multiples of 4 are: 4,8,12,16,20,24…

Multiples of 6 are: 6,12,18,24….

The common multiples for 4 and 6 are 12,24,36,48…and so on. The least common multiple in that lot would be 12. Let us now try to find out the LCM of 24 and 15.

LCM of 24 and 15
LCM of 24 and 15 = 2 × 2 × 2 × 3 × 5 = 120

LCM of Two Numbers

Suppose there are two numbers, 8 and 12, whose LCM we need to find. Let us write the multiples of these two numbers.

8 = 16, 24, 32, 40, 48, 56, …

12 = 24, 36, 48, 60, 72, 84,…

You can see, the least common multiple or the smallest common multiple of two numbers, 8 and 12 is 24.

HCF and LCM Formula

The formula which involves both HCF and LCM is:

Product of Two numbers = (HCF of the two numbers) x (LCM of the two numbers)

Say, A and B are the two numbers, then as per the formula;
A x B = H.C.F.(A,B) x L.C.M.(A,B)
We can also write the above formula in terms of HCF and LCM, such as:
H.C.F. of Two numbers = Product of Two numbers/L.C.M of two numbers
&
L.C.M of two numbers = Product of Two numbers/H.C.F. of Two numbers

How to find HCF and LCM?

Here are the methods we can use to find the HCF and LCM of given numbers.

  1. Prime factorisation method
  2. Division method

Let us learn both methods, one by one.

Prime Factorisation for HCF

Take an example of finding the highest common factor of 144, 104 and 160.
Now let us write the prime factors of 144, 104 and 160.
144 = 2 × 2 × 2 × 2 × 3 × 3
104 = 2 × 2 × 2 × 13
160 = 2 × 2 × 2 × 2 × 2 × 5
The common factors of 144, 104 and 160 are 2 × 2 × 2 = 8
Therefore, HCF (144, 104, 160) = 8

Division method to find the HCF (Shortcut method)

Steps to find the HCF of any given numbers;

1) Larger number/ Smaller Number
2) The divisor of the above step / Remainder
3) The divisor of step 2 / remainder. Keep doing this step till R = 0(Zero).
4) The last step’s divisor will be HCF.

The above steps can also be used to find the HCF of more than 3 numbers.
Example: Find the HCF of 144 and 160 by division method.
Since 160>144, so the dividend will be 160 and the divisor will be 144.
By using the division method, we get:
HCF by Long division method

Hence, we can see here 16 is the highest number which divides 160 and 144.
Therefore, HCF (144, 160) = 16

LCM By Prime Factorisation

To calculate the LCM of two numbers 60 and 45. Out of other ways, one way to find the LCM of given numbers is as below:

  • List the prime factors of each number first.
    60 = 2 × 2 x 3 × 5
    45 = 3 × 3 × 5
  • Then multiply each factor the most number of times it occurs in any number.

If the same multiple occurs more than once in both the given numbers, then multiply the factor the most number of times it occurs.
The occurrence of Numbers in the above example:
2: two times
3: two times
5: one time
LCM = 2 × 2 x 3 × 3 × 5 = 180

LCM by Division Method

Let us see with the same example, which we used to find the LCM using prime factorisation.
Solve LCM of (60,45) by division method.
LCM by long division method

Therefore, LCM of 60 and 45 = 2 × 2 x 3 × 3 × 5 = 180
At BYJU’S you can also learn, Prime Factorization Of Hcf And Lcm.

HCF and LCM Questions

Example 1: Find the Highest Common Factor of 25, 35 and 45.
Solution: Given, three numbers as 25, 35 and 45.

We know, 25 = 5 × 5

35 = 5 × 7

45 = 5 × 9

From the above expression, we can say 5 is the only common factor for all the three numbers.

Therefore, 5 is the HCF of 25, 35 and 45.

Example 2: Find the Least Common Multiple of 36 and 44.

Solution:  Given, two numbers 36 and 44. Let us find out the LCM, by division method.
HCF and LCM Questions
Therefore, LCM(36, 44) = 2 × 2 × 3 × 3 × 11 = 396

Example 3: What is the L.C.M. of 25, 30, 35 and 40?

Solution: L.C.M. of 25, 30, 35 and 40

Let us find LCM by prime factorisation.

Prime factorisation of 25 = 5 x 5 = 52

Prime factorisation of 30 = 2 x 3 x 5 

Prime factorisation of 35 = 5 x 7

Prime factorisation of 40 = 2 x 2 x 2 x 5 = 23 x 5

Thus, 

LCM (25, 30, 35, 40) = 2 x 2 x 2 x 3 x 5 x 5 x 7 = 4200

Example 4: The HCF of two numbers is 29 & their sum is 174. What are the numbers?

Solution: Let the two numbers be 29x and 29y. 

Given, 29x + 29y = 174 

29(x + y) = 174 

x + y = 174/29 = 6

Since x and y are co-primes, therefore, possible combinations would be (1,5), (2,4), (3,3)

For (1,5): 29 x = 29 x 1 and 29 y = 29 (5) = 145

Therefore, the required numbers are 29 and 145.

Practice Problems

  1. Find the HCF of 43, 91 and 183.
  2. Which is the greatest number of four digits which is divisible by 15, 25, 40 and 75?
  3. Three numbers are in the ratio of 3 : 4 : 5 and their L.C.M. is 2400. What is the value of HCF?
  4. What is the LCM of 24, 36 and 40?

5. The H.C.F. of two numbers is 11 and their L.C.M. is 7700. If one of the numbers is 275, then find the other number.

Frequently Asked Questions – FAQs

What is HCF with an example?

HCF or Highest Common Factor of two or more numbers is the greatest factor that divides the numbers. For example, 2 is the HCF of 4 and 6.

What is LCM with an example?

LCM or Least common multiple is the smallest number which is divisible by two or more given numbers. For example, LCM of 2 & 3 is 6.

What is the GCF of 24 and 36?

By prime factorisation, we know;
Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18 and 36
Factors of 24 = 1, 2, 3, 4, 6, 8, 12 and 24
HCF of (24,36) = 12

What is the formula for HCF and LCM?

Product of Two numbers = (HCF of the two numbers) x (LCM of the two numbers)

How can we find the LCM and HCF?

We can find LCM and HCF using prime factorisation and long division method

Take a quick test on HCF and LCM

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