Here we will discuss HCF and LCM with examples

## Highest Common Factor (HCF)

HCF stands for “Highest Common Factor,” which is also known as the “Greatest Common Divisor” (GCD) in some regions. It is a mathematical concept used to find the largest number that divides two or more integers without leaving a remainder.

To find the HCF of two or more numbers, you can follow these steps:

- List the factors of each number.
- Identify the common factors shared by all the numbers.
- Find the largest among these common factors.

For example, let’s find the HCF of 12 and 18:

Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 18: 1, 2, 3, 6, 9, 18

The common factors of 12 and 18 are 1, 2, 3, and 6. Among these, the largest common factor is 6. Therefore, the HCF of 12 and 18 is 6.

HCF is often used in various mathematical and real-world problems, such as simplifying fractions, finding common denominators, and determining the highest common factor among a set of numbers.

## Q Find the Highest Common Factor of 24 and 36

24 = 1,2,3,4,**6**,8,12,24

36 = 1,2,3,4,**6**,9,12,36

H.C.F = 6

Certainly! Here are a few examples of finding the HCF (Highest Common Factor) of two or more numbers in English:

## Example 1: Find the HCF of 12 and 18

Step 1: Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 18: 1, 2, 3, 6, 9, 18

Step 2: Common factors: 1, 2, 3, 6

Step 3: Largest common factor (HCF): 6

So, the HCF of 12 and 18 is 6.

## Example 2: Find the HCF of 24, 36, and 48

Step 1: Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Step 2: Common factors: 1, 2, 3, 4, 6, 12

Step 3: Largest common factor (HCF): 12

So, the HCF of 24, 36, and 48 is 12.

## Example 3: Find the HCF of 15 and 25

Step 1: Factors of 15: 1, 3, 5, 15

Factors of 25: 1, 5, 25

Step 2: Common factors: 1, 5

Step 3: Largest common factor (HCF): 5

So, the HCF of 15 and 25 is 5.

## Method Factorization:

## Euclid’s Algorithm: Find the HCF of 18 and 30

determining the HCF of 18 & 30.

12 16 20

Multiple of 6 are 12 18 24

So LCM of 4 and 6 are 12

## Q: Find the HCF of 1260 and 945.

**Step 1:** List the factors of each number.

Factors of 1260:

- 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14, 15, 18, 20, 21, 30, 35, 42, 45, 60, 63, 70, 90, 126, 210, 315, 630, 1260

Factors of 945:

- 1, 3, 5, 9, 15, 27, 45, 81, 135, 945

**Step 2:** Identify the common factors shared by both numbers.

Common factors: 1, 3, 5, 9, 15, 45

**Step 3:** Find the largest among these common factors.

The largest common factor is 45.

**Step 4:** The result is the HCF of 1260 and 945.

So, the HCF of 1260 and 945 is 45.

## Least Common Multiple (LCM)

LCM stands for “Least Common Multiple,” which is the smallest multiple that is evenly divisible by two or more integers. In other words, it is the smallest common multiple of a set of numbers.

To find the LCM of two or more numbers, you can follow these steps:

- List the multiples of each number.
- Identify the common multiples shared by all the numbers.
- Find the smallest among these common multiples.

For example, let’s find the LCM of 4 and 6:

Multiples of 4: 4, 8, 12, 16, 20, …

Multiples of 6: 6, 12, 18, 24, 30, …

The common multiples of 4 and 6 are 12, 24, 36, and so on. Among these, the smallest common multiple is 12. Therefore, the LCM of 4 and 6 is 12.

LCM is used in various mathematical operations and real-world applications, such as adding and subtracting fractions with different denominators, scheduling tasks, and working with periodic phenomena.

## Method factorization

So

The product of the H.C.F & L.C.M of two numbers is equal to the product of the number.**Example: **

Given numbers 16 and 24.

their and

then

Certainly! Let’s go through each example step by step to find the Least Common Multiple (LCM).

## Basic Examples:

### Find the LCM of 4 and 6. LCM(4, 6) = 12

### Explanation:

- List the multiples of 4: 4, 8, 12, 16, 20, …
- List the multiples of 6: 6, 12, 18, 24, 30, …
- The smallest common multiple is 12, so LCM(4, 6) = 12.

### Find the LCM of 3 and 5. LCM(3, 5) = 15

### Explanation:

- List the multiples of 3: 3, 6, 9, 12, 15, …
- List the multiples of 5: 5, 10, 15, 20, 25, …
- The smallest common multiple is 15, so LCM(3, 5) = 15.

## Intermediate Examples:

### Find the LCM of 8, 12, and 18. LCM(8, 12, 18) = 72

### Explanation:

- List the multiples of 8: 8, 16, 24, 32, 40, …
- List the multiples of 12: 12, 24, 36, 48, 60, …
- List the multiples of 18: 18, 36, 54, 72, 90, …
- The smallest common multiple is 72, so LCM(8, 12, 18) = 72.

### Find the LCM of 10 and 15. LCM(10, 15) = 30

### Explanation:

- List the multiples of 10: 10, 20, 30, 40, 50, …
- List the multiples of 15: 15, 30, 45, 60, 75, …
- The smallest common multiple is 30, so LCM(10, 15) = 30.

## Advanced Examples:

### Find the LCM of 1/4 and 2/3.

To find the LCM of fractions, first, find the LCM of the denominators and then use it as the denominator for the LCM of the fractions: LCM(4, 3) = 12 LCM(1/4, 2/3) = (1/4)*(2/3) = 2/12 = 1/6

### Find the LCM of 2.5 and 1.2.

To find the LCM of decimal numbers, you can convert them to fractions and then find the LCM: 2.5 = 5/2, and 1.2 = 6/5 LCM(5/2, 6/5) = 30/10 = 3

## Challenging Examples:

## Find the LCM of 3, 7, and 11. LCM(3, 7, 11) = 231

### Explanation:

In this case, the numbers are prime (have no common factors), so the LCM is the product of the numbers: LCM(3, 7, 11) = 3 * 7 * 11 = 231.

### Find the LCM of 36, 48, and 72. LCM(36, 48, 72) = 144

### Explanation:

- List the multiples of 36: 36, 72, 108, 144, …
- List the multiples of 48: 48, 96, 144, 192, …
- List the multiples of 72: 72, 144, 216, 288, …
- The smallest common multiple is 144, so LCM(36, 48, 72) = 144.

## Find the LCM of 15, 18, and 24. LCM(15, 18, 24) = 72

## Explanation:

- List the multiples of 15: 15, 30, 45, 60, 75, …
- List the multiples of 18: 18, 36, 54, 72, 90, …
- List the multiples of 24: 24, 48, 72, 96, 120, …
- The smallest common multiple is 72, so LCM(15, 18, 24) = 72.

### Find the LCM of 2^3, 3^2, and 5. LCM(2^3, 3^2, 5) = 360 Explanation:

- LCM(2^3, 3^2, 5) = LCM(8, 9, 5)
- List the multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, …
- List the multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, …
- List the multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, …
- The smallest common multiple is 360, so LCM(2^3, 3^2, 5) = 360.

Now you are able to solve HCF and LCM. any query contact us.