Properties of LCM and HCF With Detailed Explanation

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HCF and LCM are considered to be one of the most important and useful topics in Mathematics to solve problems related to time and work, time and distance, pipes and cisterns, etc. When two or more numbers have a common factor, their greatest common factor is their highest common factor. It is also termed HCF. Alternatively, the LCM is the smallest common multiple of these two values. The division method and the prime factorization method are the two main methods for finding H.C.F. and L.C.M. To better understand the Highest Common Factor and the Lowest Common Multiple, one must first comprehend the terms multiples and factors.

Table of Content

What are Factors and Multiples?
What is HCF?
What is LCM?
Properties of LCM and HCF
Things to Remember
Sample Questions

Key Takeaways: HCF, LCM, Factors, Multiples, Greatest Common Factor, Prime Factorization, Remainder, Least Common Divisor, Co-Prime Numbers, Fractions

What are Factors and Multiples?

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Before moving on to HCF and LCM, one must be clear about the concepts of Factors and Multiples. Given below are the definitions of Factors and Multiples.

Factor: The numbers that divide any given number completely, without leaving a remainder, are called factors. To put it another way, these are the exact divisors of the numbers.

Let us understand the factor with the help of an example:

We have a number 10. All the possible factors of 10 are 10, 5, 2, and 1 because these numbers divide 10 completely without leaving any remainder.

Multiple: A number's multiples are the numbers obtained by multiplying them with any other number.

As an example, the first 5 multiples of 4 are 4, 8, 12, 16, 20.

Factors and Multiples

Factors and Multiples

What is HCF?

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HCF is the highest common factor. The highest common factor between two or more numbers is the largest common factor of the numbers. In simple words in the largest common number dividing the two numbers. This factor is also referred to as the 'Greatest Common Factor.' Let us understand the Highest Common factor with the help of an example.

We have two numbers 8 and 20,

Factors of 8 include 8, 4, 2, 1

Factors of 20 are 20, 10, 5, 4, 2, 1

The highest common factor between 8 and 20 is 4.

Highest Common Factor (HCF)

Highest Common Factor (HCF)

What is LCM?

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LCM is the least common multiple. The least common multiple is the smallest number that is completely divided by two or more numbers. LCM is also known as the 'Least Common Divisor'. Let us understand the Lowest Common multiple with the help of an example.

We have two numbers 16 and 12,

We can write 16 = 2 × 2 × 2 × 2

And we can write 12 = 2 × 2 × 3

The lowest common multiple of 16 and 12 is 48.

Least Common Multiple (LCM)

Least Common Multiple (LCM)

The division method and prime factorization method can be used to find the Lowest Common Multiple and Highest Common Factor of two or more numbers. The division approach, on the other hand, is thought to be the quickest way to find the Highest Common Factor and Lowest Common Multiple

Read More: Relation Between HCF and LCM

Properties of LCM and HCF

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The properties of the lowest common multiple and highest common factor are explained as follows:

Property 1: The product of any 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

This property is valid for two natural numbers only, we can not use it for more than two numbers.

Lets us understand the property with an example.

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.

The HCF of co-prime numbers is given as 1. As a result, the LCM of the provided co-prime numbers equals the product of the numbers.

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

Lets us understand the 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.

Read More: Prime Numbers

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

Lets us understand the property with an example.

We have two numbers 20 and 15.

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

Ans the factors of 15 include: 15, 5, 3, 1

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

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

Lets us understand the 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.

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

Lets us understand the 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

Read More: Real Numbers

Things to Remember

The Highest Common Factor of two or more numbers is the greatest common factor of the numbers.
The LCM, or the lowest common multiple, on the other hand, is the smallest common multiple of these two values.
The numbers that divide any given number completely, without leaving a remainder, are called factors. A number's multiples are the numbers obtained by multiplying them with any other number.
Two major approaches for finding H.C.F. and L.C.M. are the division method and the prime factorization method.
The product of any two natural numbers' LCM and HCF is equal to the product of the given numbers. This property is valid for two numbers only, we can not use it for more than two numbers.
Co-prime numbers have an HCF of 1. As a result, the LCM of the provided co-prime numbers equals the product of the numbers.
The highest common factor of any given number will never be bigger than any of the numbers.
The lowest common multiple of any given numbers will never be lesser than any of the numbers.

Sample Questions

Ques. Prove that the product of LCM and HCF of any given numbers is equal to the product of the numbers themselves. (3 Marks)

Ans. Lets us take two numbers 8 and 24

The HCF of 10 and 24 is 2

For 10 and 24, the LCM is 120

HCF × LCM = 2 × 120

= 240

The product of 10 and 24, 8 x 20 = 240

We know, the product of LCM and HCF = Product of the given numbers.

Hence, The product of any two natural numbers' LCM and HCF is equal to the product of the given numbers.

Ques. Two co-prime numbers 7 and 9 are given. Using these numbers verify that the LCM of Co-prime Numbers is the same as the product of the numbers. (3 Marks)

Ans. We have two numbers 7 and 9.

For 7 and 9, the LCM is 63

The product of the numbers 7 and 9 is 63.

Hence, the LCM of Co-prime Numbers is the same as the product of the numbers.

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

Ans. 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

Ques. The LCM of the two co-prime numbers is 35. What will be the product of the given numbers? (3 Marks)

Ans. According to the properties of HCF and LCM of numbers,

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

So, the product of the given numbers will be 35.

Ques. Write down the factors and multiples of 32? (2 Marks)

Ans. The factors and multiples of 32 are as follows:

Factors of 32 = 32, 16, 8, 4, 2, 1

Multiples of 32 = 32, 64, 128, 256....and so on.

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

Ans. 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

Ques. Calculate the HCF of 48 and 56 (3 Marks)

Ans. To calculate the Highest Common Factor, write down the factors of 48 and 56.

The factors that make 48 = 48, 24, 12, 8, 6, 4, 2,1

And factors that make 56 = 56, 28, 14, 8, 7, 4, 2, 1

So, for 48 and 56. the HCF is 8.

Ques. The HCF of two numbers is 24 and one of the two numbers is 16. State true or false with appropriate reason (3 Marks)

Ans. The statement is false as the properties of the HCF states that the highest common factor of any given number will never be bigger than any of the numbers. Here one given number is 16 and HCF is 24 which is greater than the number.

We can understand this property of HCF with the help of an example.

Consider two numbers 20 and 15.

Factors of 20 = 20, 10, 5, 4, 2, 1

Factors of 15 = 15, 5, 3, and 1

Therfore, the HCF of 15 and 20 will be 5 which is lesser than both the numbers 20 and 15.

Related Links:

Difference Between Whole Numbers and Natural Numbers	Number System
Polynomials	Factoring Formula
Real Numbers Formula	Prime Number Formula
Finding Square Roots through Prime Factorization	Cube Roots
Factorisation of Polynomials	MCQs for Real Numbers

Properties of LCM and HCF With Detailed Explanation

What are Factors and Multiples?

What is HCF?

What is LCM?

Properties of LCM and HCF

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

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

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

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

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

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

2.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

3.
Find:
Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

4.
At a birthday party, children are being served orange juice in conical cups, as shown in the figure.

Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.
On the basis of the above information, answer the following questions :

5.
If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

6.
Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

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Properties of LCM and HCF With Detailed Explanation

What are Factors and Multiples?

What is HCF?

What is LCM?

Properties of LCM and HCF

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

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

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

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

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

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

2. Find: If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

3.Find:Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

4.At a birthday party, children are being served orange juice in conical cups, as shown in the figure. Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.On the basis of the above information, answer the following questions :

5.If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

6.Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

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1.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

2.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

3.
Find:
Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

4.
At a birthday party, children are being served orange juice in conical cups, as shown in the figure.

Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.
On the basis of the above information, answer the following questions :

5.
If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

6.
Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).