HCF and LCM

Maths is considered a creative subject that involves the application of principles and solving problems. Be it daily life calculation or entrance exam preparation, mathematics has always been an important subject to cover for students. For some reason, students who do not understand the concepts of Maths develop an unnecessary fear for the subject that hurdles their growth towards success. In this regard, Vedantu, an emerging online educational platform for students helps them to master learning with ease for all subjects including Maths. 

LCM and HCF are important topics of Mathematical calculation that keep evolving and remain important in the syllabus for Maths students. It is therefore important to build a strong foundation right from the beginning by learning the very essence of LCM and HCF. 

What does HCF mean?

• The largest common factor of all the given numbers is known as the Highest Common Factor of the numbers.

• The highest number can be divided exactly into two or more numbers without any remainders.

• It is also known as the Greatest Common Divisor (GCD).

• The easiest way to find the HCF of two or more given numbers is to create a factor tree.

Here are a few highest common factor examples : 

(Image will be uploaded soon)

The above picture shows how you can calculate the H.C.F. of 90 and 30 using the prime factorization method.

How is HCF calculated?

There are three methods of how to find the highest common factor of any two or more given numbers:

1.  Factorization Method

2. Prime Factorization Method

3. Division Method

How to calculate LCM and HCF?

This is how to calculate LCM & HCF:-

How to find the HCF of 3 Numbers or How to find the Highest Common Factor?

We can find the HCF of 3 numbers either by Prime Factorisation Method or by Division Method. However, the steps for finding the highest common factor remain the same as above.

Here are a few highest common factor examples of how to find the highest common factor.

1. Find the HCF of 3 numbers 15, 30 and 90 using the Prime Factorization method.

Solution:

15 = 5 \[\times\] 3 \[\times\] 1

30 = 5 \[\times\] 3 \[\times\] 2 \[\times\] 1

90 = 3 \[\times\] 3 \[\times\] 2 \[\times\] 5 \[\times\] 1

The common factors here are 1,3,5.

Therefore, the highest common factor of the numbers,15,30 and 90 is 5×3×1=15.

How is LCM Calculated?

LCM by Listing Multiples -

Step 1: You need to list the multiples of each number until at least one of the multiples appears on all the lists.

Step 2: Now find the smallest number that is on all of the lists

Step 3: This number is the Least Common Multiple.

Example: Let’s find the LCM of (6,7,21)

Write down the multiples of 6:  6, 12, 18, 24, 30, 36, 42, 48, 54, 60

Write down the multiples of 7:  7, 14, 21, 28, 35, 42, 56, 63

Write down the multiples of 21: 21, 42, 63

Now you need to find the smallest number that is present on all of the lists. So the LCM(6, 7, 21) is 42.

How to find LCM by Prime Factorization?

Step 1: Firstly, find all the prime factors of each given number.

Step 2: Now, list down all the prime numbers found, as many times as these numbers occur most often for anyone given number.

Step 3: In step 3, you need to multiply the list of prime factors together to find the LCM.

Step 4: The LCM (a,b) can be calculated by finding the prime factorization of both the numbers a and b. We can do the same process for the LCM of more than 2 numbers.

For example, Let’s find the LCM (12,30) we find:

First, find the prime factorization of 12 = 2 \[\times\] 2 \[\times\] 3

Second, the prime factorization of 30 = 2 \[\times\] 3 \[\times\] 5

Using all prime numbers we have found as often as each occurs most often we take 2 \[\times\] 2 \[\times\] 3 \[\times\] 5 = 60

Therefore, the LCM (12,30) = 60.

How to find LCM by Prime Factorization using the Concept of Exponents?

Step 1: Firstly, you need to find all the prime factors of each of the given numbers and write all the prime factors in exponent form.

Step 2: Now you need to list all the prime numbers found, using the highest exponent found for each of them.

Step 3: Lastly, multiply the list of prime factors you have with exponents together to find the Least common multiple.

For example: Find the LCM(12,18,30)

List down the prime factors of 12 = 2 \[\times\] 2 \[\times\] 3 = 22 \[\times\] 31

List down the prime factors of 18 = 2 \[\times\] 3 \[\times\] 3 = 21 \[\times\] 32

List down the prime factors of 30 = 2 \[\times\] 3 \[\times\] 5 = 21 \[\times\] 31 \[\times\] 51

You need to list all the prime numbers found, the number of times as they occur most often for anyone given number and you need to multiply them together to find the Least common multiple.

After multiplying, 2 \[\times\] 2 \[\times\] 3 \[\times\] 3 \[\times\] 5 = 180

Using the concept of exponents instead, multiply together each of the prime numbers with the highest power

In exponential form, 22 \[\times\] 32 \[\times\] 51 = 180

So, the LCM(12,18,30) = 180.

Where are HCF and LCM used?

Questions to be solved:

1. Find out the LCM of(24,300)

Prime factors of 24 = 2 \[\times\] 2 \[\times\] 2 \[\times\] 3 = 23 \[\times\] 31

Prime factors of 300 = 2 \[\times\] 2 \[\times\] 3 \[\times\] 5 \[\times\] 5 = 22 \[\times\] 31 \[\times\] 52

List all the prime numbers found, as many times as they occur most often for anyone given number and multiply them together to find the LCM

2 \[\times\] 2 \[\times\] 2 \[\times\] 3 \[\times\] 5 \[\times\] 5 = 600.

Using exponents instead, multiply together each of the prime numbers with the highest power

23 \[\times\] 31 \[\times\] 52 = 600.

So LCM (24,300) = 600.

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