GCD Made Easy: Math Problems & Solutions

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Hey math enthusiasts! Ever found yourself scratching your head over finding the greatest common divisor (GCD)? Well, you're in the right place! Today, we're going to break down how to find the GCD of a few sets of numbers, making it super easy to understand. We'll be working through several examples to make sure you get the hang of it. Ready to dive in? Let's go!

What is the Greatest Common Divisor (GCD)?

So, what exactly is the greatest common divisor? In simple terms, the GCD of two or more numbers is the largest number that divides evenly into all of them. Think of it as the biggest number that can be a factor of each number without leaving a remainder. For instance, the GCD of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 without any leftovers. Understanding the concept of GCD is fundamental in number theory and has applications in various fields, from simplifying fractions to solving more complex mathematical problems. Keep in mind that finding the GCD is a skill that comes with practice, so don't be discouraged if you don't get it right away. The more you work through examples, the easier it will become. Knowing the GCD can help you with a lot of different mathematical problems, so it's a super useful thing to know. We are going to go through a variety of problems so that you can see how it works in action. Plus, we'll explain each step in a way that's easy to follow. By the end of this article, you'll be a GCD pro!

Now, there are a few ways to find the GCD, but we'll focus on the method of listing the factors and then finding the largest common one. There's also the Euclidean algorithm, which is another awesome method, but we will focus on the listing of factors method for now. This approach is more intuitive, especially when you're just starting out. It's all about breaking down the numbers into their factors, finding the ones they share, and then picking the biggest one. So, grab a pen and paper, and let's start solving some problems!

Finding GCD: Step-by-Step Examples

Alright, guys, let’s get our hands dirty with some examples! We'll tackle each set of numbers step by step to make sure you understand how to find the GCD like a pro. Keep in mind, practice makes perfect! So, the more you work through these examples, the better you’ll get.

a) GCD of 24 and 36

Let’s begin with the numbers 24 and 36. Here’s how we'll find their GCD:

  1. List the factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
  2. List the factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
  3. Identify the common factors: 1, 2, 3, 4, 6, 12
  4. Determine the greatest common factor: 12

Therefore, the GCD of 24 and 36 is 12. See? Not so hard, right?

b) GCD of 45 and 54

Next up, we have 45 and 54:

  1. List the factors of 45: 1, 3, 5, 9, 15, 45
  2. List the factors of 54: 1, 2, 3, 6, 9, 18, 27, 54
  3. Identify the common factors: 1, 3, 9
  4. Determine the greatest common factor: 9

So, the GCD of 45 and 54 is 9. Awesome!

c) GCD of 27 and 72

Let’s find the GCD of 27 and 72:

  1. List the factors of 27: 1, 3, 9, 27
  2. List the factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
  3. Identify the common factors: 1, 3, 9
  4. Determine the greatest common factor: 9

Therefore, the GCD of 27 and 72 is 9. Easy peasy!

d) GCD of 18, 36 and 48

Now, let's try finding the GCD of three numbers: 18, 36, and 48.

  1. List the factors of 18: 1, 2, 3, 6, 9, 18
  2. List the factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
  3. List the factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
  4. Identify the common factors: 1, 2, 3, 6
  5. Determine the greatest common factor: 6

So, the GCD of 18, 36, and 48 is 6. You see how it works? Even with three numbers, it is the same process.

e) GCD of 125, 200 and 375

Let's keep the momentum going and calculate the GCD of 125, 200, and 375:

  1. List the factors of 125: 1, 5, 25, 125
  2. List the factors of 200: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 200
  3. List the factors of 375: 1, 3, 5, 15, 25, 75, 125, 375
  4. Identify the common factors: 1, 5, 25
  5. Determine the greatest common factor: 25

Thus, the GCD of 125, 200, and 375 is 25.

f) GCD of 144, 156 and 192

Okay, let's find the GCD of 144, 156, and 192:

  1. List the factors of 144: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144
  2. List the factors of 156: 1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156
  3. List the factors of 192: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 192
  4. Identify the common factors: 1, 2, 3, 4, 6, 12
  5. Determine the greatest common factor: 12

So, the GCD of 144, 156, and 192 is 12.

g) GCD of 180, 216 and 288

Now, let's calculate the GCD of 180, 216, and 288:

  1. List the factors of 180: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180
  2. List the factors of 216: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216
  3. List the factors of 288: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 96, 144, 288
  4. Identify the common factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 36
  5. Determine the greatest common factor: 36

Therefore, the GCD of 180, 216, and 288 is 36.

h) GCD of 1200, 1800 and 2000

Time to find the GCD of 1200, 1800, and 2000:

  1. List the factors of 1200: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 25, 30, 40, 50, 60, 75, 100, 120, 150, 200, 300, 400, 600, 1200
  2. List the factors of 1800: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 25, 30, 36, 40, 45, 50, 60, 75, 90, 100, 150, 180, 200, 225, 300, 360, 450, 600, 900, 1800
  3. List the factors of 2000: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 400, 500, 1000, 2000
  4. Identify the common factors: 1, 2, 4, 5, 10, 20, 25, 50, 100, 200
  5. Determine the greatest common factor: 200

So, the GCD of 1200, 1800, and 2000 is 200. We're on a roll!

i) GCD of 150, 450 and 600

Lastly, let's find the GCD of 150, 450, and 600:

  1. List the factors of 150: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150
  2. List the factors of 450: 1, 2, 3, 5, 6, 9, 10, 15, 18, 25, 30, 45, 50, 75, 90, 150, 225, 450
  3. List the factors of 600: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 25, 30, 40, 50, 60, 75, 100, 120, 150, 200, 300, 600
  4. Identify the common factors: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150
  5. Determine the greatest common factor: 150

Therefore, the GCD of 150, 450, and 600 is 150.

Conclusion: Mastering the GCD

And there you have it, folks! We've covered how to find the greatest common divisor through a series of examples. Remember, the key is to list the factors of each number, identify the common ones, and then pick the largest. Keep practicing, and you'll become a GCD master in no time! This method is a great starting point, and it’s super useful for understanding the concept. So, keep up the great work, and don't hesitate to revisit these examples whenever you need a refresher. You've got this!

Also, a good tip is that when dealing with larger numbers, it might be beneficial to use a calculator to find the factors. This helps you to avoid missing any factors, which can be time-consuming. However, understanding the core principles is what is important. With time, you'll be able to quickly find the GCD of any set of numbers! Remember, practice makes perfect. Keep practicing, and you'll be acing these problems in no time. If you have any questions or want to try some more examples, feel free to ask. Keep learning and keep exploring the amazing world of mathematics! You've got all the tools you need to succeed. Keep up the good work, and happy calculating!