GCD & LCM: Analyzing A Student's Math Claim

Hey guys! Let's dive into a common math problem that often trips up students: finding the Greatest Common Divisor (GCD) and Least Common Multiple (LCM). We'll analyze a student's claim about the GCD and LCM of 72 and 120, figuring out whether they're on the right track or need a little more practice. This is super important because understanding GCD and LCM is the foundation for a bunch of other math concepts. Let's get started and break down this math problem!

The Student's Claim: GCD, LCM, and Math Errors

Alright, so here's the deal: a student is claiming that the GCD of 72 and 120 is 8, and the LCM is 960. Our job is to figure out if this is correct. Before we jump to conclusions, let's remember what GCD and LCM actually mean. The GCD is the largest number that divides both given numbers without any remainder, whereas the LCM is the smallest number that both given numbers divide into. It's like finding the biggest common factor and the smallest common multiple, respectively. This student's claim gives us a chance to brush up on these fundamental concepts. We will use two simple methods to resolve this problem, the prime factorization method and the division method. These methods will help to ensure that we will understand the process of solving such questions.

Now, the student's statement is the starting point. It's crucial to check if the numbers 8 and 960 really fit the definitions of GCD and LCM for 72 and 120. If we're working on a math test, we'll want to take our time to determine the correct answers. We can also use this as an opportunity to review the underlying math concepts and ensure that they are understood. Because the student's answers may not be entirely wrong, but may contain some errors, we need to take a step-by-step approach to resolve the problem. Also, this type of problem helps us to understand and grasp the concepts of math better, so that we can approach and solve similar problems in the future without any problems. So, let's start with checking the student's GCD claim.

Finding the Correct GCD: Prime Factorization and Division Method

To find the correct GCD of 72 and 120, let's use two common methods: prime factorization and the division method. Prime factorization involves breaking down each number into a product of prime numbers. For 72, the prime factorization is 2 x 2 x 2 x 3 x 3 (or 2³ x 3²). For 120, it's 2 x 2 x 2 x 3 x 5 (or 2³ x 3 x 5). To find the GCD, we look for the common prime factors and their lowest powers. In this case, both numbers share 2³ and 3. Multiplying these together (2³ x 3), we get 24. Therefore, the actual GCD of 72 and 120 is 24, not 8, as the student claimed. Thus, the student's GCD claim is incorrect.

Then, we can use the division method to find the correct GCD. In this method, we divide the larger number (120) by the smaller number (72). We will then take the divisor (72) and divide it by the remainder (48). Then, we will take the divisor (48) and divide it by the remainder (24). Since the remainder is zero, the GCD of the two numbers is 24. This shows us that the student's answer is wrong. Using both methods can help us check and ensure our understanding of this concept. It also gives us a clear understanding that the student is incorrect because the correct GCD is 24. This part is critical because it highlights the importance of precise calculations. So it's very important to ensure all steps are correct.

This simple process of prime factorization, and division method, helps us double-check our work. It also builds up a systematic approach to tackle any GCD problem. Guys, this is how you become a math master!

Determining the Correct LCM: Prime Factorization and Division Method

Okay, now let's tackle the LCM. Remember, the LCM is the smallest number that both 72 and 120 can divide into evenly. Using the prime factorizations we found earlier, 72 = 2³ x 3² and 120 = 2³ x 3 x 5. To find the LCM, we take the highest power of each prime factor present in either number. So, we take 2³ (from both), 3² (from 72), and 5 (from 120). Multiplying these together (2³ x 3² x 5), we get 360. That means the correct LCM of 72 and 120 is 360, not 960, as the student suggested. The student's LCM claim is also incorrect.

And now let's resolve this problem by the division method. The first step involves dividing the numbers by their common factors, so that the two numbers 72 and 120 will be divided by 2. This will result in 36 and 60. Then divide both 36 and 60 by 2, which gives you 18 and 30. Then, divide the two numbers by 2, which gives 9 and 15. Finally, divide 9 and 15 by 3, which gives you 3 and 5. Then, we can multiply the divisors and the remaining numbers. In our case, it will be 2 x 2 x 2 x 3 x 3 x 5. This equals 360. In this case, 360 is the LCM of the numbers. Both of these methods help us double-check our results to ensure that we understand this concept.

Evaluating the Student's Claim

So, to wrap it up: The student claimed the GCD was 8 and the LCM was 960. However, the correct GCD is 24, and the correct LCM is 360. Therefore, the statement that evaluates the student's claim is: "The student is incorrect because the correct GCD is 24 and the LCM is 360." The student might have made a calculation error, or perhaps they misunderstood the concept. This exercise underscores the importance of precise calculations and a solid grasp of fundamental mathematical concepts like GCD and LCM. Now, it is very important to remember the methods we have learned to resolve this problem.

Conclusion: Mastering GCD and LCM

Alright, guys, we've broken down this problem and learned a lot about GCD and LCM. Always remember to double-check your work, use prime factorization or the division method, and practice consistently. These concepts are fundamental in math and will help you tackle more complex problems down the road. Keep practicing, and you'll become math rockstars in no time! So, keep learning, keep questioning, and keep having fun with math! If you have any questions or want to try another problem, just let me know!