## Introduction

The Least Common Multiple (LCM) of a set of numbers is the smallest number that can be evenly divided by each number in the set. Finding the LCM is an essential concept in math, especially when we're working with fractions or solving problems that involve synchronization of different cycles. Let's focus on how to find the LCM of 45, 60, and 75 using different methods.

## What is the Least Common Multiple (LCM)?

The LCM is the smallest number that is a multiple of two or more numbers. For instance, when we need to find the LCM of 45, 60, and 75, we are looking for the smallest number that all three of these numbers can divide into without leaving a remainder. This is useful in many real-world applications, such as finding common denominators in fractions or planning events to avoid scheduling conflicts.

## Ways to Calculate the LCM

There are several methods to calculate the LCM of a set of numbers. The most common methods are:

- Division Method
- Listing Method
- Prime Factorization Method

Each of these methods provides a systematic way of finding the LCM, and we can choose any based on our preference or the complexity of the numbers involved.

## Prime Factorization Method

To find the LCM of 45, 60, and 75 using the prime factorization method, we follow these steps:

- Break down each number into its prime factors.
- Identify the highest power of each prime number involved.
- Multiply these highest powers together to get the LCM.

Let's break down each number:

45 = 3^2 * 5

60 = 2^2 * 3 * 5

75 = 3 * 5^2

The highest powers of the prime factors are:

- 2^2
- 3^2
- 5^2

Now, multiply these together:

LCM = 2^2 * 3^2 * 5^2 = 4 * 9 * 25 = 900

So, the LCM of 45, 60, and 75 is **900**.

## Division Method

In the division method, we follow these steps:

- Write the numbers in a row.
- Divide by the smallest prime number that can divide at least one of the numbers.
- Write the quotient directly below the numbers not divisible.
- Continue this process until we get 1 in all rows.

45, 60, 75

Divide by 2:

45, 30, 75

Divide by 2:

45, 15, 75

Divide by 3:

15, 5, 25

Divide by 3:

5, 5, 25

Divide by 5:

1, 1, 5

Divide by 5:

1, 1, 1

Now, multiply all the divisors we used:

2 * 2 * 3 * 3 * 5 * 5 = 900

So, the LCM of 45, 60, and 75 is **900**.

## Listing Method

To find the LCM by listing method, we follow these steps:

- List some multiples of each number.
- Identify the smallest common multiple.

Let's list some multiples:

Multiples of 45: 45, 90, 135, 180, 225, 270, 315, 360, 405, 450, 495, 540, 585, 630, 675, 720, 765, 810, 855, 900

Multiples of 60: 60, 120, 180, 240, 300, 360, 420, 480, 540, 600, 660, 720, 780, 840, 900

Multiples of 75: 75, 150, 225, 300, 375, 450, 525, 600, 675, 750, 825, 900

The smallest common multiple is 900.

So, the LCM of 45, 60, and 75 is **900**.

## Formula to Calculate LCM

The formula to calculate the LCM of two or more numbers is:

LCM = (Number1 * Number2) / GCD(Number1, Number2)

For three numbers, it can be generalized as:

LCM(45, 60, 75) = (LCM(45, 60) * 75) / GCD(LCM(45, 60), 75)

We can find LCM(45, 60) first and then use this formula to find the LCM of all three numbers.

## Conclusion

Finding the LCM of a set of numbers like 45, 60, and 75 is quite simple once we understand the methods involved. The prime factorization method, division method, and listing method all help us get the same result, which, in this case, is 900.

## FAQs

**Q: What is the LCM?**

A: The Least Common Multiple is the smallest number that is a multiple of two or more numbers.

**Q: Why do we need to find the LCM?**

A: We need the LCM for various reasons, such as solving fraction problems or synchronizing schedules.

**Q: Which method is the best to find the LCM?**

A: The best method depends on the numbers involved and personal preference. The prime factorization method is often straightforward, but the division and listing methods also provide accurate results.

**Q: Can the LCM be smaller than any of the numbers?**

A: No, the LCM is always equal to or larger than the largest number in the set.