lcm and hcf

LL Certainly! LCM (Least Common Multiple) and GCD (Greatest Common Divisor), also known as HCF (Highest Common Factor), are fundamental concepts in number theory.

LCM (Least Common Multiple): 

The LCM of two or more numbers is the smallest multiple that is divisible by each of the given numbers. To find the LCM of two numbers, you can use the following steps:

1. List the prime factors of each number.

2. Take the highest power of each prime factor from the lists.

3. Multiply these highest powers together.

For example, let's find the LCM of 12 and 18:

- Prime factors of 12: 2^2 * 3

- Prime factors of 18: 2 * 3^2

The highest powers are 2^2 and 3^2. Therefore, LCM (12, 18) = 2^2 * 3^2 = 4 * 9 = 36.

GCD (Greatest Common Divisor) or HCF (Highest Common Factor):

The GCD or HCF of two or more numbers is the largest number that divides each of the given numbers. To find the GCD of two numbers, you can use the following steps:

1. List the prime factors of each number.

2. Take the lowest power of each prime factor from the lists.

3. Multiply these lowest powers together.

For example, let's find the GCD of 24 and 36:

- Prime factors of 24: 2^3 * 3

- Prime factors of 36: 2^2 * 3^2

The lowest powers are 2^2 and 3. Therefore, GCD (24, 36) = 2^2 * 3 = 12.

These concepts are important in various mathematical operations and problem-solving scenarios, including fractions, algebraic simplification, and more.

Sure, let's work through an example of finding the LCM and GCD (HCF) of two numbers: 24 and 36.

Example: Finding LCM and GCD of 24 and 36

Step 1: Prime Factorization

First, let's find the prime factorization of each number:

- Prime factors of 24: 2^3 * 3^1 (since 24 = 2 * 2 * 2 * 3)

- Prime factors of 36: 2^2 * 3^2 (since 36 = 2 * 2 * 3 * 3)

Step 2: Finding LCM

To find the LCM, we take the highest powers of each prime factor:

 LCM = 2^3 * 3^2 = 8 * 9 = 72

So, the LCM of 24 and 36 is 72.

Step 3: Finding GCD (HCF)

To find the GCD, we take the lowest powers of each prime factor:

 GCD = 2^2 * 3^1 = 4 * 3 = 12

So, the GCD (HCF) of 24 and 36 is 12.

Summary:

 LCM of 24 and 36: 72

GCD (HCF) of 24 and 36: 12

In this example, the LCM is the smallest multiple that is divisible by both 24 and 36, while the GCD (HCF) is the largest number that divides both 24 and 36.