HCF and LCM Formula With Trick for Competitive Exams

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 Definition

The full form of HCF in Maths is 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 Definition

The full form of LCM in Maths is 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 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.

Tricks to Find LCM easily

Example 1: Find LCM of 2,4,8,16.

Choose the largest number. In this example, the largest number is 16. Check whether 16 is divisible by all other remaining numbers. 16 is divisible by 2, 4, 8. Hence, the LCM is 16.

Example 2: Find the LCM of 2,3,7,21.

Choose the largest number. The largest number is 21. Check whether 21 is divisible by all other remaining numbers. 21 is divisible by 3 and 7 but not by 2. So multiply 21 and 2. The result is 42. Now, check whether 42 is divisible by 2, 3, 7. Yes, 42 is divisible. Hence, the LCM is 42. 

Example 3: Find the LCM of 3,5,15,30

Choose the largest number. The largest number is 30. Check whether 30 is divisible by all other remaining numbers. 30 is divisible by 3, 5, and 15. Hence, the LCM is 30.

Example 4: Find the LCM of 8,24,48,96

Choose the largest number. The largest number is 96. Check whether 96 is divisible by all other remaining numbers. 96 is divisible by 8, 24, and 48. Hence, the LCM is 96.

Example 5: Find the LCM of 12,36,60,108

Choose the largest number. The largest number is 108. Check whether 108 is divisible by all other remaining numbers. 108 is divisible by 12, 36, and 60. Hence, the LCM is 108.

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

NOTE-  The above relation between H.C.F and L.C.M is not valid for the product of numbers greater than 2. It is only valid for the product 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 factorization method
2. Division method

Let us learn both methods, one by one.

HCF by Prime Factorization Method

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

HCF by Division 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.

Frequently Asked Questions on HCF and LCM