Finding Numbers With A Least Common Multiple Of 36

Jul 11, 2025 by ADMIN 51 views

Understanding Least Common Multiple Finding Possible Number Pairs

The least common multiple (LCM) of two numbers is a fundamental concept in number theory. It's the smallest positive integer that is divisible by both numbers. This article delves into the problem of finding two numbers given their LCM. Specifically, we'll address the question: The least common multiple of two numbers is 36. Which could be the two numbers? This involves understanding the definition of LCM and applying it to a multiple-choice question to identify the correct pair of numbers. This article aims to provide a comprehensive explanation of how to determine the LCM and then use that knowledge to solve the presented problem, offering a step-by-step approach that can be applied to similar mathematical challenges.

Understanding the Least Common Multiple (LCM)

To tackle the given problem effectively, a clear understanding of the least common multiple is crucial. The LCM of two numbers is the smallest positive integer that is divisible by both numbers without leaving a remainder. In simpler terms, it’s the smallest number that both numbers can divide into evenly. Let's break down the concept with an example before applying it to the problem at hand. Consider the numbers 4 and 6. Multiples of 4 are 4, 8, 12, 16, 20, 24, and so on. Multiples of 6 are 6, 12, 18, 24, 30, and so on. The common multiples of 4 and 6 are 12, 24, and so on. The smallest of these common multiples is 12, therefore, the LCM of 4 and 6 is 12. This basic understanding is key to solving problems involving LCMs, including identifying number pairs that yield a specific LCM. The process involves listing multiples of each number until a common multiple is found. The smallest common multiple is the LCM. In more complex scenarios, prime factorization can be used to efficiently determine the LCM, but for smaller numbers, listing multiples is often a straightforward approach. Understanding the LCM is not only essential for solving mathematical problems but also has practical applications in everyday situations, such as scheduling events or dividing quantities. With a solid grasp of the concept, we can now apply it to solve the problem of finding two numbers with an LCM of 36.

Problem Breakdown: LCM of 36

The problem states that the least common multiple (LCM) of two numbers is 36. Our task is to identify which pair of numbers from the given options could have 36 as their LCM. This requires a systematic approach of checking each option to see if the LCM of the pair is indeed 36. To solve this, we'll go through each option, list the multiples of each number in the pair, and identify the smallest common multiple. If the smallest common multiple is 36, then that pair is a potential solution. If not, we move on to the next option. This method ensures that we find the correct pair that satisfies the condition of having 36 as their LCM. The key is to accurately list the multiples and carefully identify the smallest common one. Let's delve into the options one by one, applying this method to determine the correct answer. We'll start with option A and proceed through the list, ensuring we thoroughly evaluate each pair of numbers. This process not only helps us find the solution but also reinforces our understanding of LCM and how to calculate it.

Evaluating Option A: 9 and 12

Let's evaluate option A, which presents the numbers 9 and 12. To determine if the least common multiple (LCM) of 9 and 12 is 36, we need to list the multiples of each number and find the smallest multiple they have in common. The multiples of 9 are: 9, 18, 27, 36, 45, and so on. The multiples of 12 are: 12, 24, 36, 48, and so on. By comparing these lists, we can see that the smallest multiple that both 9 and 12 share is 36. Therefore, the LCM of 9 and 12 is indeed 36. Since we have found a pair of numbers that satisfies the condition given in the problem, this suggests that option A could be the correct answer. However, to be certain, we should still evaluate the other options to ensure there isn't another pair that also has an LCM of 36. This step is crucial to avoid selecting a potentially incorrect answer and to reinforce the understanding of the problem-solving process. Although option A appears to be correct, let's proceed with evaluating the remaining options to confirm our findings.

Evaluating Option B: 4 and 6

Now, let's evaluate option B, which includes the numbers 4 and 6. To find the least common multiple (LCM) of 4 and 6, we again list the multiples of each number and identify the smallest common multiple. The multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, and so on. The multiples of 6 are: 6, 12, 18, 24, 30, 36, and so on. Comparing these multiples, we find that the smallest number that appears in both lists is 12. Thus, the LCM of 4 and 6 is 12, not 36. This means that option B does not satisfy the condition given in the problem, where the LCM of the two numbers must be 36. Since option B does not meet the criteria, we can eliminate it as a possible solution. This step reinforces the importance of accurately calculating the LCM for each option to ensure the correct answer is selected. With option B ruled out, we proceed to evaluate option C, continuing our systematic approach to problem-solving.

Evaluating Option C: 3 and 8

Next, we consider option C, which gives us the numbers 3 and 8. To determine the least common multiple (LCM) of 3 and 8, we list the multiples of each number and look for the smallest multiple they share. The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, and so on. The multiples of 8 are: 8, 16, 24, 32, 40, and so on. Comparing the lists, we find that the smallest multiple common to both 3 and 8 is 24. Therefore, the LCM of 3 and 8 is 24, not 36. This indicates that option C does not meet the requirement of the problem, as the LCM of the numbers must be 36. Consequently, we can eliminate option C from our potential solutions. This step further narrows down our choices and highlights the importance of careful calculation in determining the LCM. With option C eliminated, we now move on to the final option, D, to complete our evaluation.

Evaluating Option D: 3 and 12

Finally, let's evaluate option D, which presents the numbers 3 and 12. To find the least common multiple (LCM) of 3 and 12, we list the multiples of each number and identify the smallest multiple that appears in both lists. The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, and so on. The multiples of 12 are: 12, 24, 36, 48, and so on. By comparing these lists, we can see that the smallest multiple that both 3 and 12 share is 12. Therefore, the LCM of 3 and 12 is 12, not 36. This means that option D does not satisfy the condition of the problem, which requires the LCM of the two numbers to be 36. Since option D does not meet the criteria, we can eliminate it as a possible solution. With all options now evaluated, we can confidently conclude which pair of numbers has an LCM of 36.

Conclusion: Identifying the Correct Pair

After evaluating all the options, we can now confidently identify the correct pair of numbers whose least common multiple (LCM) is 36. We systematically examined each option, listing the multiples of the numbers and determining their LCM. Option A, with the numbers 9 and 12, was found to have an LCM of 36. Options B, C, and D, with number pairs 4 and 6, 3 and 8, and 3 and 12 respectively, had LCMs of 12, 24, and 12, none of which matched the required LCM of 36. Therefore, the correct answer is option A: 9 and 12. This exercise demonstrates the importance of understanding the concept of LCM and applying it methodically to problem-solving. By listing multiples and comparing them, we can accurately determine the LCM of two numbers and identify pairs that meet specific LCM criteria. The process also reinforces the need for careful calculation and attention to detail in mathematical problem-solving.

By working through this problem, we've not only found the solution but also deepened our understanding of LCM and its application in identifying number pairs. This knowledge can be applied to a variety of mathematical problems and real-world scenarios where LCM is a relevant concept.