LCM And GCD: Calculate For Multiple Number Sets
Hey everyone! Today, we're going to dive into finding the Least Common Multiple (LCM) and Greatest Common Divisor (GCD) for a bunch of different number sets. It might sound a bit intimidating, but trust me, we'll break it down step by step so it's super easy to follow. So, grab your pencils and let's get started!
1: 108 and 72
Let's kick things off with finding the LCM and GCD of 108 and 72. Understanding these concepts is super useful in many areas, from simplifying fractions to solving real-world problems. So, pay close attention, guys!
First, we need to find the prime factorization of each number. For 108, it's 2^2 * 3^3. For 72, it's 2^3 * 3^2. Now, to find the GCD, we take the lowest power of each common prime factor. In this case, the common prime factors are 2 and 3. The lowest power of 2 is 2^2, and the lowest power of 3 is 3^2. So, the GCD is 2^2 * 3^2 = 4 * 9 = 36.
Next, to find the LCM, we take the highest power of each prime factor present in either number. That's 2^3 * 3^3 = 8 * 27 = 216. So, the LCM of 108 and 72 is 216. See? Not so hard when you break it down!
To recap, the GCD (Greatest Common Divisor) is the largest number that divides both 108 and 72 without leaving a remainder. The LCM (Least Common Multiple) is the smallest number that both 108 and 72 divide into evenly. Understanding how to find these is super handy in math. Keep practicing, and you'll become a pro in no time!
2: 18 and 45
Now, let's tackle the LCM and GCD of 18 and 45. This is another great opportunity to practice our prime factorization skills and solidify our understanding of GCD and LCM.
First, we break down each number into its prime factors. 18 can be written as 2 * 3^2, and 45 can be written as 3^2 * 5. To find the GCD, we look for the common prime factors and take the lowest power of each. The only common prime factor here is 3, and the lowest power is 3^2. So, the GCD of 18 and 45 is 3^2 = 9.
For the LCM, we take the highest power of each prime factor present in either number. That gives us 2 * 3^2 * 5 = 2 * 9 * 5 = 90. Therefore, the LCM of 18 and 45 is 90. Remember, the LCM is the smallest multiple that both numbers divide into evenly.
Understanding these concepts is super important, and with a bit of practice, you'll find it becomes second nature. Keep at it, and you'll be amazed at how quickly you improve!
3: 27 and 16
Okay, guys, let's move on to finding the LCM and GCD of 27 and 16. This one is interesting because the numbers don't share any common prime factors (except 1, of course!).
Let's start with prime factorization. 27 is 3^3, and 16 is 2^4. Since they don't have any common prime factors, their GCD is 1. That's right, when two numbers have no common factors other than 1, they are called relatively prime, and their GCD is always 1.
Now, for the LCM, we simply multiply the two numbers together: 27 * 16 = 432. So, the LCM of 27 and 16 is 432. This makes sense because, without any common factors, the smallest multiple they both share is their product.
This example highlights an important point: when numbers are relatively prime, finding the GCD and LCM is straightforward. The GCD is 1, and the LCM is the product of the numbers. Keep this in mind as you tackle similar problems in the future!
4: 36, 20, and 90
Alright, let's level up and find the LCM and GCD of 36, 20, and 90. Now we're dealing with three numbers, but the process is still the same, just a bit more involved.
First, we find the prime factorization of each number: 36 = 2^2 * 3^2, 20 = 2^2 * 5, and 90 = 2 * 3^2 * 5. To find the GCD, we look for the common prime factors in all three numbers and take the lowest power of each. The common prime factors are 2. The lowest power of 2 is 2^1 = 2. Therefore, the GCD of 36, 20, and 90 is 2.
For the LCM, we take the highest power of each prime factor present in any of the numbers. That's 2^2 * 3^2 * 5 = 4 * 9 * 5 = 180. So, the LCM of 36, 20, and 90 is 180. Remember, this is the smallest number that all three numbers divide into evenly.
When working with more than two numbers, the key is to stay organized and make sure you consider all the prime factors. Keep practicing, and you'll become a pro at finding the LCM and GCD of any set of numbers!
5: 45, 54, and 60
Next up, we're finding the LCM and GCD of 45, 54, and 60. Let's keep sharpening those prime factorization skills!
First, we break down each number: 45 = 3^2 * 5, 54 = 2 * 3^3, and 60 = 2^2 * 3 * 5. To find the GCD, we look for common prime factors in all three numbers. The only common prime factor is 3, and the lowest power is 3^1 = 3. So, the GCD of 45, 54, and 60 is 3.
Now, for the LCM, we take the highest power of each prime factor present in any of the numbers: 2^2 * 3^3 * 5 = 4 * 27 * 5 = 540. Therefore, the LCM of 45, 54, and 60 is 540.
It's worth noting that the LCM can sometimes be a fairly large number, especially when the numbers have several different prime factors. Don't be intimidated by large numbers – just stick to the process, and you'll get there!
6: 28, 35, 63
Let's keep the ball rolling and find the LCM and GCD of 28, 35, and 63. Keep practicing, guys; you're doing great!
First, we find the prime factorizations: 28 = 2^2 * 7, 35 = 5 * 7, and 63 = 3^2 * 7. The GCD is the product of the lowest powers of the common prime factors. The only prime factor common to all three numbers is 7 (7^1 in each case). So, the GCD is 7.
Next, we find the LCM by taking the highest power of each prime factor that appears in any of the factorizations: 2^2 * 3^2 * 5 * 7 = 4 * 9 * 5 * 7 = 1260. Thus, the LCM of 28, 35, and 63 is 1260.
Make sure to double-check your work to avoid errors, especially when working with multiple numbers. Accuracy is key in math!
7: 20, 30, and 50
Okay, let's find the LCM and GCD of 20, 30, and 50. By now, you should be getting pretty comfortable with this process.
First, let's break down each number: 20 = 2^2 * 5, 30 = 2 * 3 * 5, and 50 = 2 * 5^2. To find the GCD, we identify the common prime factors and take the lowest power of each. The common prime factors are 2 and 5. The lowest power of 2 is 2^1 = 2, and the lowest power of 5 is 5^1 = 5. So, the GCD of 20, 30, and 50 is 2 * 5 = 10.
Now, for the LCM, we take the highest power of each prime factor present in any of the numbers: 2^2 * 3 * 5^2 = 4 * 3 * 25 = 300. Therefore, the LCM of 20, 30, and 50 is 300.
Remember, the GCD is the largest number that divides all the given numbers without leaving a remainder, while the LCM is the smallest number that is a multiple of all the given numbers. Keep these definitions in mind as you practice.
8: 720, 600, and 540
Now, let's tackle some bigger numbers! We're finding the LCM and GCD of 720, 600, and 540. Don't worry, the process is the same, just with larger prime factorizations.
First, we find the prime factorization of each number: 720 = 2^4 * 3^2 * 5, 600 = 2^3 * 3 * 5^2, and 540 = 2^2 * 3^3 * 5. To find the GCD, we look for the common prime factors and take the lowest power of each: 2^2 * 3^1 * 5^1 = 4 * 3 * 5 = 60. Therefore, the GCD of 720, 600, and 540 is 60.
For the LCM, we take the highest power of each prime factor present in any of the numbers: 2^4 * 3^3 * 5^2 = 16 * 27 * 25 = 10800. So, the LCM of 720, 600, and 540 is 10800.
When dealing with larger numbers, it can be helpful to use a calculator to assist with the multiplication. Just make sure you understand the underlying process!
9: 220, 275, 1925
Alright, let's keep going! We're finding the LCM and GCD of 220, 275, and 1925. Are you feeling like a pro yet?
Let's start with the prime factorizations: 220 = 2^2 * 5 * 11, 275 = 5^2 * 11, and 1925 = 5^2 * 7 * 11. The GCD is the product of the lowest powers of the common prime factors. The common prime factors are 5 and 11. So the GCD will be 5^1 * 11^1 = 55.
Next, we find the LCM by taking the highest power of each prime factor that appears in any of the factorizations: 2^2 * 5^2 * 7 * 11 = 4 * 25 * 7 * 11 = 7700. Thus, the LCM of 220, 275, and 1925 is 7700.
Keep in mind that if a prime factor appears in only one of the numbers, you still include it in the LCM calculation. This ensures that the LCM is divisible by all the original numbers.
10: 605, 1925, and 2695
Last but not least, let's find the LCM and GCD of 605, 1925, and 2695. You've made it to the final problem—great job!
First, we find the prime factorization of each number: 605 = 5 * 11^2, 1925 = 5^2 * 7 * 11, and 2695 = 5 * 7^2 * 11. To find the GCD, we look for the common prime factors and take the lowest power of each: 5^1 * 11^1 = 5 * 11 = 55. Therefore, the GCD of 605, 1925, and 2695 is 55.
For the LCM, we take the highest power of each prime factor present in any of the numbers: 5^2 * 7^2 * 11^2 = 25 * 49 * 121 = 150025. So, the LCM of 605, 1925, and 2695 is 150025.
And there you have it! You've successfully found the LCM and GCD for all ten sets of numbers. You're well on your way to mastering these important concepts.
Remember, practice makes perfect. The more you work with LCM and GCD, the easier it will become. Keep up the great work, and don't be afraid to ask for help if you get stuck. You got this!