Math will not be a tough subject, especially when you understand the concept and relationship between one another. Before learning the relationship between H.C.F and L.C.M let us find what H.C.F and L.C.M is. HCF and LCM are the two terms that stand for the highest common factor and least common multiple respectively. Check out this page to know the definition of H.C.F and L.C.M, Relationship between HCF and L.C.M. In this article, we will focus on the relation between H.C.F and L.C.M with solved examples.

Also Refer:

### HCF and LCM – Definitions

**Highest Common Factor:** HCF is the greatest factor of two numbers or more than two numbers which divides the number exactly with no remainder.

**Example:** Find the Highest Common Factors of two numbers 9 and 12.

Solution:

Prime factorization of 9: 3 × 3

Prime factorization of 12: 2 × 2 × 3

3 is the only common factor for both the numbers 9 and 12.

**Least Common Multiple:** LCM of two numbers or more than two numbers is the smallest number which is divisible by given numbers exactly.

**Example:** Find the LCM of 9 and 12.

Solution:

Multiples of 9 are 9, 18, 27, 36, 45,…

Multiples of 12 are 12, 24, 36, 48,…

The least common multiple of 9 and 12 is 36.

### HCF and LCM Relation

The relation between HCF and LCM provides an easy way to solve the problems.

**1. HCF and LCM of Positive Integers**

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

HCF × LCM = product of the two numbers

Let A and B be two natural numbers

HCF(A & B) × LCM (A & B) = A × B

**Example:** Show that LCM(6, 15) × HCF(6, 15) = product(6, 15)

Solution:

6 = 2 × **3**

15 = **3** x 5

LCM of 6 and 15 = 30

HCF of 6 and 15 = 3

LCM (6, 15) × HCF (6, 15) = 30 × 3 = 90

Product of 6 and 15 = 6 × 15 = 90

Thus, the LCM (6, 15) × HCF (6, 15)=Product(6, 15) = 90

**2. HCF and LCM of Co-Prime Numbers**

The L.C.M of the given co-prime numbers is equal to the product of the numbers since the H.C.G of ci-prime is 1.

So, LCM of Co-prime Numbers = Product Of The Numbers

**Example:** 11 and 13 are the two co-prime numbers.

Solution:

factors of 11: **1** × 11

Factors of 13: **1** × 13

LCM of 11 and 13 is 143

and HCF of 11 and 13 is 1.

Hence it is proved that L.C.M of the given co-prime numbers is equal to the product of the numbers since the H.C.G of ci-prime is 1.

**3. H.C.F. and L.C.M. of Fractions**

LCM of fractions = LCM of Numerators/HCF of Denominators

HCF of fractions = HCF of Numerators/LCM of Denominators

**Example:** Find the LCM of the fractions 2/3, 4/5, 3/4

Solution:

LCM of fractions = LCM of Numerators/HCF of Denominators

LCM of fractions = LCM (2,4,3)/HCF(3,5,4)

**Example:** Find the HCF of the fractions 3/5, 6/11, 9/20

Solution:

HCF of fractions HCF of Numerators/LCM of Denominators

HCF of fractions = HCF (3,6,9)/LCM (5,11,20)=3/220

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### Relationship between HCF and LCM Examples

**Example 1.**

Find the HCF and LCM of the following numbers

1. 117, 221

2. 145, 232

**Solution:**

1. 117, 221

Prime factorization of 117 = 3 × 3 × 13

Prime factorization of 221 = 13 × 17

HCF = 13

LCM is the product of coprime numbers.

LCM = 3 × 3 × 13 × 17 = 1989

2. 145, 232

Prime factorization of 145 = 5 × 29

232 = 2 × 2 × 2 × 29

So, the required HCF = 29

Required LCM = 2 × 2 × 2 × 5 × 29 = 1160

**Example 2.**

The LCM and HCF of the two numbers are 180 and 6. If one of the numbers is 36, find the other number.

**Solution:**

Given,

LCM of two numbers = 180

HCF of two numbers = 6

One of the numbers = 36

We know that

Product of two numbers = product of HCF and LCM

36 × other number = 6 × 180

Other number = (6 × 180)/36 = 30

Thus the other number is 30.

**Example 3.**

13 and 17 are the two co-prime numbers. By using the given numbers prove that,

LCM of given co-prime Numbers = Product of the given Numbers

**Solution:**

Factors of 13 are 1 × 13

Factors of 17 are 1 × 17

HCF of 13 and 17 = 1

LCM of 13 and 17 = 221

Product of 13 and 17 = 13 × 17 = 221

Thus LCM of co-prime numbers = Product of the numbers

**Example 4.**

The HCF of the two numbers is 1 and their LCM is 35. If one of the numbers is 7, find the other number.

**Solution:**

Given,

The HCF of the two numbers is 1 and their LCM is 35.

One of the numbers = 7

HCF × LCM = Product of two numbers

1 × 35 = 7 × second number

35 = 7 × second number

second number = 35/7

second number = 5

Thus the other number is 5.

**Example 5.**

Find the HCF and LCM of 12 and 18.

**Solution:**

The prime factorization of 12 is 2 ×** 2 × 3**

The prime factorization of 18 is **2 × 3** × 3

The H.C.F of 12 and 18 is 6.

LCM of 12 and 18

Multiples of 12 are 12, 24, **36**,..

Multiples of 18 are 18, **36**, …

Thus the LCM of 12 and 18 is 36.

Therefore the HCF and LCM of 12 and 18 are 6 and 36.

### FAQs on HCF and LCM Relation

**1. What is the formula of LCM and HCF?**

HCF = HCF of Numerators/LCM of denominators.

LCM = LCM of Numerators/ HCF of Denominators.

**2. What is the relation between HCF and LCM of two numbers?**

The product of the H.C.F. and L.C.M. of any two numbers is always equal to the product of those two numbers.

**3. What are the 3 methods of HCF?**

There are three methods of finding H.C.F. of two or more numbers.

1. Factorization Method.

2. Prime Factorization Method.

3. Division Method.