LCM of 120 and 144

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:

1. Division Method
2. Listing Method
3. 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:

1. Find the prime factors of each number.
2. List out all the prime factors.
3. For each different prime factor, take the highest power of that prime factor.
4. 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:

1. Write the numbers side by side.
2. Divide by the smallest prime number that can divide at least one of the numbers.
3. 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:

1. List some multiples of each number.
2. 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.