**Introduction**

Hey there! Let's dive into the world of math and talk about something called the **Least Common Multiple (LCM)**. Today, we're going to learn how to find the LCM of the numbers **120** and **144**. Understanding the LCM can be super helpful in many areas, like solving problems with fractions, arranging schedules, and more.

**What Is LCM?**

The **Least Common Multiple (LCM)** of two numbers is the smallest number that both original numbers can divide into without leaving a remainder. In other words, it's the smallest number that's a multiple of both the given numbers. For instance, the LCM of 120 and 144 is the smallest number that both 120 and 144 can divide into exactly.

**Ways To Calculate The LCM**

There are a few different methods we can use to find the LCM:

- Division Method
- Listing Method
- Prime Factorization Method

**Prime Factorization Method**

The **Prime Factorization Method** involves breaking down each number into its prime factors. Here’s how we do it:

- Find the prime factors of each number.
- List out all the prime factors.
- For each different prime factor, take the highest power of that prime factor.
- Multiply these together to get the LCM.

Let’s do this with 120 and 144.

Prime factorization of 120:

120 = 2^3 * 3 * 5

Prime factorization of 144:

144 = 2^4 * 3^2

Now, take the highest powers of all prime factors:

2^4 (from 144)

3^2 (from 144)

5^1 (from 120)

Multiplying these together gives:

LCM = 2^4 * 3^2 * 5^1 = 16 * 9 * 5 = 720

So, the LCM of 120 and 144 is 720.

**Division Method**

In the **Division Method**, we divide the numbers by their common prime factors until only 1s are left. Here’s how it works:

- Write the numbers side by side.
- Divide by the smallest prime number that can divide at least one of the numbers.
- Write the quotients below and repeat the process until all numbers are 1.

Let's do this for 120 and 144:

Step 1:

2 | 120 144

2 | 60 72

2 | 30 36

2 | 15 18

3 | 15 9

3 | 5 3

5 | 5 1

1 1

Now, multiply all the divisors:

LCM = 2 * 2 * 2 * 2 * 3 * 3 * 5 = 720

Again, the LCM is 720.

**Listing Method**

The **Listing Method** involves listing the multiples of each number until we find the smallest common one. Here’s how we do it:

- List some multiples of each number.
- Find the smallest multiple that appears in both lists.

Multiples of 120:

120, 240, 360, 480, 600, 720, 840, …

Multiples of 144:

144, 288, 432, 576, 720, 864, …

The smallest common multiple is 720. So, the LCM of 120 and 144 is 720.

**Formula For Calculating LCM**

There is a simple formula to calculate the LCM using the Greatest Common Divisor (GCD):

LCM(a, b) = (a * b) / GCD(a, b)

First, let’s find the GCD of 120 and 144:

Using prime factorizations:

GCD = 2^3 * 3 = 24

Now, apply the formula:

LCM(120, 144) = (120 * 144) / 24 = 720

So, using the formula, the LCM is 720.

**Conclusion**

We’ve explored different ways to find the **LCM of 120 and 144**. Whether using the **Prime Factorization Method**, the **Division Method**, or the **Listing Method**, we always end up with the same result: **720**. These methods give us the flexibility to choose the one that works best for us in different situations.

**FAQs**

**What Is The LCM Used For?**

The **LCM** is useful for solving problems where we need to find a common multiple, such as in scheduling, adding fractions, and more.

**Can We Use The LCM Formula For All Numbers?**

Yes, the LCM formula works for all pairs of positive integers.

**Is The LCM Always Bigger Than The Given Numbers?**

Usually, the **LCM** is equal to or larger than the largest of the given numbers.

**What If One Number Is A Multiple Of The Other?**

If one number is a multiple of the other, the **LCM** is the larger number.

**Does The LCM Change With Negative Numbers?**

We generally use the absolute values of numbers to find the **LCM**, so it doesn’t change.