Relation Between HCF and LCM

To understand the relationship between HCF and LCM we need to understand the basic concepts used in defining them. These two concepts are closely related to each other and are used in solving many math problems. In this article we will define the relation between HCF and LCM in detail along with related concepts and examples.

Highest Common Factor (HCF)

The highest common factor of a number is defined as the largest number that divides two more numbers completely without leaving a remainder. For example, HCF of 32 and 64 is 32 because 32 is the only largest number that divides both 32 and 64 completely.

We can also define HCF as the largest number among the common factors of two or more given numbers is called the Highest Common Factor (HCF). It is also known as Greatest Common Divisor (GCD)

Steps to find HCF:
HCF by common factors
• List the factors of the given numbers.
• Find the common factors.
• The greatest among the common factors is HCF.

For example: Consider the numbers 16, 24, 64.
The factors of 16 are: 1, 2, 4, 8, 16
The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24
The factors of 64 are: 1, 2, 4, 8, 16, 32, 64
The common factors are: 1, 2, 4, 8
The highest common factor: HCF = 8

Least Common Multiple (LCM)

The least common divisor of a number is defined as the smallest number that is the multiple of all the numbers present in a group. For example, the LCM of 2, 3, 4, and 6 is 12.

Steps to find LCM:
LCM by common multiples
• List the multiples of the given numbers.
• Find the common multiples.
• The smallest among the common multiples is the LCM of the given numbers.

For example: LCM of numbers 6 and 8.
Multiples of 6 = 6, 12, 18, 24, 30 and so on
Multiples of 8 = 8, 16, 24, 32, 40 and so on
LCM of 6 and 8 is 24.

Relationship Between HCF and LCM 

Understanding the relationship between HCF and LCM helps you to solve problems easily. Here are the relations between HCF and LCM of two numbers:

1) HCF and LCM of Natural Numbers:
The product of the HCF and LCM of two natural numbers is the same as the product of these numbers. Let’s consider P and Q as two natural numbers then the LCM of (P,Q) multiplied by HCF of (P, Q) gives the product of the numbers. 

I.e., LCM (P, Q) x HCF (P, Q) = P x Q

For example, HCF of 16 and 28 is 4 whereas LCM of 16 and 28 is 112 

 

2) HCF and LCM of Co-prime Numbers:

The LCM of co-prime numbers is equal to the product of the numbers as the HCF of co-prime numbers is 1. For example, if A and B is two co-prime number then LCM of (A, B) = product of (A, B).

I.e., LCM (A, B) = A x B

3). HCF and LCM of Fraction:

The HCM and LCM of fractions is defined by HCF and LCM of numerators of fractions divided by the LCM or HCF of the denominator of the fractions. 

For example, if a/b and c/d are two fractions then their HCF and LCM are as follows:

HCF =    HCF of Numerators (a, c)LCM of Denominators (b, d)

LCM =  LCM of Numerators (a, c)HCF of Denominators (b, d)

 

Solved Examples

Example 1: Find the HCF of 36 and 48.
Solution:  24022021055123621839331 
36 = 2 × 2 × 3 × 3
48 = 2 × 2 × 2 × 2 × 3
The product of common factors: 2 × 2 × 3 = 12. So, the HCF for the numbers 36 and 48 is 12.

Example 2: Find the HCF of 12, 15 and 21 using prime factorisation method.
Solution:  21226331315551321771
12 = 2 × 2 × 3
15 = 3 × 5
21 = 3 × 7
HCF of 12, 15 and 21 = 3