# LCM and HCF

## LCM Definition

In mathematics, the Least Common Multiple, abbreviated as **LCM**, is a fundamental concept. The LCM of two numbers, such as *a* and *b*, is represented as `LCM(a, b)`

. It is defined as the smallest positive integer that is divisible by both *a* and *b*.

For instance, consider the positive integers 4 and 6:

- The multiples of 4 are: 4, 8, 12, 16, 20, 24, and so on.
- The multiples of 6 are: 6, 12, 18, 24, and so forth.

The common multiples of 4 and 6 include: 12, 24, 36, 48, and so on. Among these, the smallest common multiple is 12. Hence, the LCM of 4 and 6 is 12.

Now, let's determine the LCM of 24 and 15:

- The multiples of 24 are: 24, 48, 72, 96, 120, and so on.
- The multiples of 15 are: 15, 30, 45, 60, 75, 90, 105, 120, and so forth.

The common multiples of 24 and 15 include: 120, 240, 360, etc. Among these, the smallest common multiple is 120. Therefore, the LCM of 24 and 15 is 120.

**Also Check: Full Form of HCF**

## LCM of Two Numbers

### Finding the LCM of 8 and 12

Let's try to find the Least Common Multiple (LCM) of two numbers, 8 and 12. We will look at the multiples of these two numbers to determine their LCM.

- The multiples of 8 are: 8, 16, 24, 32, 40, 48, 56, and so on.
- The multiples of 12 are: 12, 24, 36, 48, 60, 72, 84, and so forth.

As you can see, the smallest common multiple of the numbers 8 and 12 is 24. Therefore, the LCM of 8 and 12 is 24.

### Understanding HCF

In mathematics, HCF stands for Highest Common Factor. The largest positive integer that divides two or more positive integers without leaving a remainder is called the Greatest Common Divisor (GCD) or HCF.

Consider the numbers 8 and 12. The maximum number that can divide both 8 and 12 is 4. Therefore, the HCF of 8 and 12 is 4.

**Also Chek: How to find HCF**

### Example: Finding the HCF of 24 and 36

To find the HCF of 24 and 36, we use prime factorization:

- 24 = 2 × 2 × 2 × 3
- 36 = 2 × 2 × 3 × 3

By factoring the numbers, we see that the common factors are 2 × 2 × 3. Therefore, the HCF of 24 and 36 is:

**HCF(24, 36) = 2 × 2 × 3 = 12**

## HCF and LCM Formula

### Combining HCF and LCM

The Highest Common Factor (HCF) and the Least Common Multiple (LCM) are related through a fundamental formula in mathematics. This relationship is given by the following formula:

**A × B = HCF(A, B) × LCM(A, B)**

Here, *A* and *B* are two numbers, HCF(A, B) is the Highest Common Factor of A and B, and LCM(A, B) is the Least Common Multiple of A and B.

### Formulas in Terms of HCF and LCM

Using the relationship between HCF and LCM, we can derive the following formulas:

**Finding HCF**

The HCF of two numbers can be calculated as:

**HCF of Two Numbers = (Product of Two Numbers) / (LCM of Two Numbers)**

**Finding LCM**

The LCM of two numbers can be calculated as:

**LCM of Two Numbers = (Product of Two Numbers) / (HCF of Two Numbers)**

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## Frequently Asked Questions on LCM and HCF

The rule for finding the Least Common Multiple (LCM) and Highest Common Factor (HCF) of two or more numbers is:

- Write each number as the product of its prime factors
- For LCM, multiply each prime factor by the highest power it occurs in any number
- For HCF, multiply the common prime factors with the lowest power they occur

The rule for finding the Highest Common Factor (HCF) of two or more numbers is:

- Write each number as the product of its prime factors
- List the common prime factors
- Multiply the common factors using the lowest power they occur

To find the Least Common Multiple (LCM) quickly, use the prime factorization method:

- Write each number as the product of its prime factors
- Multiply each prime factor by the highest power it occurs in any number

To find the Highest Common Factor (HCF) quickly, use the prime factorization method:

- Write each number as the product of its prime factors
- List the common prime factors
- Multiply the common factors using the lowest power they occur

The list method to find the Highest Common Factor (HCF) involves:

- List the factors of each number
- Find the common factors between the numbers
- The largest common factor is the HCF