Properties of HCF and LCM with Solved Examples

Given their great utility in resolving issues pertaining to time and effort, time and distance, pipes and cisterns, etc., HCF and LCM are regarded as two of the most central topics in mathematics. Understanding L.C.M. and H.C.F. of two or more numbers speeds up problem solving and cuts down on computation time. Numerous mathematical puzzles, such as figuring out the largest tile size or the largest tape to measure the area, can be solved with HCF. Numerous mathematical issues pertaining to racetracks and traffic lights can be resolved with the help of LCM. In computer science, LCM is also helpful for creating encrypted messages with the use of cryptography.

To know more about HCF and LCM, see the link mentioned below:

What is HCF?

Higher Common Factor is referred to as HCF.

• A non-zero number that represents the largest of all of their common factors is the H.C.F. of two or more numbers.
• Another name for H.C.F. is the Greatest Common Divisor (G.C.D.).

For example: Find the HCF of 18 and 21.

All the factors of 18 are 1, 2, 3, 6, 9, and 18	All the factors of 21 are 1, 3, 7, and 21
1 ×18=18 2 × 9=18 3 × 6=18	1 × 21=21 7 × 3=21

It is evident that the greatest element that 18 and 21 have in common is 3.

The greatest number that divides all of the specified integers in this case is 3. Thus, the HCF of 18 and 21 is 3.

All You Need to Know About HCF and LCM

What is LCM?

Lowest or Least Common Multiple is referred to as LCM. The smallest positive integer that is divisible by each of the provided integers is known as the LCM of two or more numbers.

Let’s take this as an example: 

Find the LCM of 16 and 20. The formula for the LCM of 16 and 20 is 2 × 2 × 2 × 2 × 5 = 80. In this case, the LCM of 16 and 20 is 80.

Also Read: What is the Full Form of HCF and LCM?

Formula of HCF and LCM

The prime factorization of the numbers can be taken into consideration while writing the LCM and HCF formulas.

LCM = Product of each prime factor’s highest power that is present in the numbers.

HCF = The product of each common prime factor’s lowest power in the given numbers.

For Example:

Let’s take two numbers, 9 and 25.

The prime factorisation of 9 and 25 can be written as:

9 = 3 × 3 = 3²

25 = 5 × 5 = 5²

Now,

LCM = 3² × 5² = 9 × 25 = 225

HCF = 1

This method of finding LCM and HCF of numbers is known as the Prime factorisation Method.

Properties of HCF and LCM

You can use the properties of HCF and LCM to divide things into smaller portions and to distribute any number of sets of objects equally into the largest grouping possible. As we study the characteristics of HCF and LCM, let’s take a look at some of the intriguing links between them. Here are a few of the most significant characteristics of HCF and LCM:

Property 1: The product of the two natural numbers LCM and HCF is equal to the product of the given numbers.

We can write it as, LCM × HCF = Product of the given numbers

We cannot apply this property for more than two natural numbers; it is only valid for two numbers.

Let’s use an example to further grasp the property.

We have two numbers 16 and 20.

For 16 and 20, the HCF is 4

Taking 16 and 20 together, the LCM is 80

HCF × LCM = 4 × 80

= 320

The product of 16 and 20, 16 × 20 = 320

Hence, LCM × HCF = Product of the given numbers.

Property 2: Co-prime numbers have an HCF of 1.

For co-prime numbers, the HCF is 1. Consequently, the product of the numbers is equal to the LCM of the given co-prime numbers.

We can write it like LCM of co-prime numbers = Product of the given numbers

Let’s understand the same property with an example.

We have two numbers, 16 and 15.

The LCM of 16 and 15 is 240

The product of 16 and 15,

16 × 15 = 240

Hence, LCM of co-prime numbers = Product of the given numbers.

Property 3: The highest common factor of any given number will never be bigger than any of the numbers.

Let’s understand the given property with an example.

We have two numbers, 20 and 15.

The factors of 20 include: 20, 10, 5, 4, 2, 1

The factors of 15 include: 15, 5, 3, 1

So, for 15 and 20, the HCF is 5. which is less than both the numbers 20 and 15.

Property 4: The lowest common multiple of any given number will never be less than any of the numbers.

Let’s understand the given property with an example.

We have two numbers, 25 and 15.

For 25 and 15, the LCM is 75

So, the LCM of 25 and 15 is 75. which is greater than both the numbers 25 and 15.

Also Read: Multiplication and Division Word Problems

Property 5: Finding HCF and LCM for the fraction values.

Consider the following two fractions: (a/b) and (c/d). The generalised formula for calculating the LCM and HCF of (a/b) and (c/d) is given below:

HCF of (a/b) and (c/d) = HCF (a, c) / LCM (b, d)

LCM of (a/b) and (c/d) = LCM (a, c) / HCF (b, d)

This means that:

H.C.F = HCF of Numerators / LCM of Denominators

L.C.M = LCM of Numerators / HCF of Denominators

Let’s understand the given property with an example.

We have two fractions, 4/7 and 12/ 5.

The HCF of 4/7 and 12/ 5 = HCF (4, 12) / LCM (7, 5)

= 4/35

The LCM of 4/7 and 12/ 5 = LCM (4, 12) / HCF (7, 5)

= 12/1 = 12

Solved Examples

1. Prove that: LCM (9 & 12) × HCF (9 & 12) = Product of 9 and 12

Solution:

9 = 3 × 3 = 3²

12 = 2 × 2 × 3 = 2² × 3

LCM of 9 and 12 = 2² × 3² = 4 × 9 = 36

HCF of 9 and 12 = 3

LCM (9 & 12) × HCF (9 & 12) = 36 × 3 = 108

Product of 9 and 12 = 9 × 12 = 108

Hence, LCM (9 & 12) × HCF (9 & 12) = 9 × 12 = 108. Proved.

2. Find the HCF of 12/25, 9/10, 18/35, 21/40.

Solution:

12 = 2 × 2 × 3

9 = 3 × 3

18 = 2 × 3 × 3

21 = 3 × 7

HCF (12, 9, 18, 21) = 3

25 = 5 × 5

10 = 2 × 5

35 = 5 × 7

40 = 2 × 2 × 2 × 5

LCM(25, 10, 35, 40) = 5 × 5 × 2 × 2 × 2 × 7 = 1400

The required HCF = HCF(12, 9, 18, 21)/LCM(25, 10, 35, 40) = 3/1400

3. Calculate the Highest Common factor of 3/7, 2/9, 12/20 (3 Marks)

Solution:

We can calculate the highest common factor by,

HCF = HCF (3, 2, 12) / LCM (7, 9, 20) 

HCF of 3, 2, 12 = 1

LCM of 7, 9, 20 = 1260

Hence, the Highest Common factor of 3/7, 2/9, 12/20 = 1/1260

4. Calculate the lowest common multiple of 16/21 and 4/9. (3 Marks)

Solution:

Using the property of HCF and LCM in fractions, 

LCM = LCM (16, 4) / HCF (21, 9) 

LCM of 16, 4 = 16

HCF of 21, 9 = 3

Hence, the Lowest Common multiple of 16/21 and 4/9 = 16/3

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