### LCM AND HCF

#### Concept of LCM and HCF:

LCM and HCF are amongst the most important quantitative topics since questions related to LCM and HCF are asked in the exam almost every year. So, here is a brief introduction to the topic along with some illustrations to help the candidates

**Least Common Multiple( LCM)**

A common multiple is one that is a multiple of 2 or more than 2 numbers.

**For example:**

**The common multiples of 2 and 3 are 6,12,18, etc.**

The Least Common Multiple of 2 numbers is the smallest positive number that is a multiple of both.

In other words, the Least Common Multiple of 2 or more numbers is the smallest number which is divisible by all the given numbers.

Multiples of 2: 2,4,6,8…

Multiples of 3: 3,6,9,12…

LCM of 2 and 3 will be 6.

**How To Find Out LCM:**

**Method 1: Prime Factorization Method**

- Factorize all numbers into their prime factors
- Make a note of all the distinct factors
- Raise each factor to the maximum power present and multiply them all

**Example:**

**To find the LCM of 136,144,168**

136 = 2^{3} x17

144 = 2^{4} x 3^{2}

168= 2^{3}x3x7 Distinct factors are 2, 17, 3 and 7.

The highest power of 2 is 4, of 3 is 2, of 17 is 1 and of 7 is 1. So, LCM = 2^{4} x 3^{2}x17 x7 = 17136

**Method 2:**

- To calculate the LCM of 4, 5 and 6, take the highest number: 6 in this case.
- Now start with the multiples of 6 and check whether they are the multiples of 4 and 5 or not.
- The first common multiple (i.e. the multiple of all 3?4,5, and 6) will be the LCM.
- You start with 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 So, 60 is the LCM as it is the first number to be divisible by 4,5 and 6.

**Highest Common Factor (HCF)**

Greatest Common divisor (GCD), also called HCF, is the largest integer that perfectly divides two or more given numbers.

**For Example:**

**The HCF of 18 and 12 is 6.**

6 is the largest number that divides both 18 and 12.

**How To Find Out HCF:**

**Method 1: Prime Factorization Method**

- Factorize all numbers into their prime factors
- Make a note of all the distinct factors present in all three numbers
- Raise each factor to the minimum power present and multiply them all

**Example: To find the HCF of 136,144, 168**

**To find the HCF of 136,144, 168**

Step 1: 136 = 2^{3} x17 144 = 2^{4} x 3^{2} 168= 2^{3}x3x7

Step 2: Distinct factors present = 2,3,7,17

Step 3: Raising each factor to the minimum present (i.e. 2^{3},3^{0},7^{0} and 17^{0}) HCF= 2^{3}=8

**Method 2: Division Method**

- To find the HCF of 2 numbers by Division Method, the higher number is divided by the lower number.
- Then the lower number is divided by the remainder obtained in the previous division.
- This remainder is divided by the next remainder and so on till the remainder is zero. The last divisor will be the HCF of the two numbers

**Example:**

**To find the HCF of 12 and 15**

**To find the HCF of 12 and 15**

15/12|R = 3 12/3|R=0

Thus, the HCF= 3

**HCF and LCM of Fractions**

HCF of fractions = HCF of numerators/ LCM of denominators

LCM of fractions = LCM of numerators/ HCF of denominators

LCM x HCF = Product of two numbers (this can be applied only for 2 numbers)

**Example 1:**

HCF of 12 and 24 = 12. LCM of 12 and 24= 24 HCF x LCM = 12×24 = product of the two numbers

This formula can be applied to any number of numbers only if all the numbers are relatively prime.

**Properties of HCF and LCM:**

- The HCF of two or more numbers is lesser than or equal to the smallest of those numbers
- The LCM of two or more numbers is greater than or equal to the greatest of those numbers
- If a number X always leaves a remainder R when divided by the numbers A,B,C.., then X= LCM(or a multiple of LCM) of A,B,C…+R