Finding The Numbers: LCM Of 36 Explained

Hey math enthusiasts! Let's dive into a fun problem. We're given that the least common multiple (LCM) of two numbers is 36, and we're tasked with figuring out which pair of numbers could possibly have this LCM. This is a classic math puzzle that helps solidify your understanding of multiples and factors. So, grab your pencils (or your favorite digital devices!), and let's get started!

Understanding the Least Common Multiple (LCM)

Before we jump into the options, let's make sure we're all on the same page about what the LCM actually is. The least common multiple of two or more numbers is the smallest positive integer that is divisible by all the numbers. Think of it this way: it's the smallest number that each of the given numbers divides into without leaving a remainder. For example, the LCM of 4 and 6 is 12 because 12 is the smallest number that both 4 and 6 divide into evenly. Understanding the concept is key to solving this type of problem. We will break down each option and see how it relates to our main keyword least common multiple. We're looking for a pair of numbers where 36 is the smallest number they both divide into. If we find that the LCM of a pair is something different than 36, then we know that's not our correct answer. This way of thinking will get us to the correct answer without too much trouble.

Now, let's think about some key aspects of finding the LCM. One easy way to find the LCM of two numbers is to list out the multiples of each number until you find the smallest multiple that they have in common. Another approach is to use prime factorization. Break down each number into its prime factors. The LCM is then the product of the highest powers of all prime factors that appear in either factorization. Let's see how these principles apply to our answer choices and how they help us find the correct answer when the least common multiple is known. For this problem, we'll go through the answer choices step by step.

Analyzing the Answer Choices

Now, let's take a look at the options one by one, keeping in mind our goal: to find the pair of numbers whose LCM is 36.

A. 9 and 12

Let's start by figuring out the LCM of 9 and 12. The multiples of 9 are: 9, 18, 27, 36, 45, ... The multiples of 12 are: 12, 24, 36, 48, ... Ah-ha! The smallest number that appears in both lists is 36. This means the least common multiple of 9 and 12 is indeed 36. So, option A is a potential answer. But we need to check the other options just to be sure that this is the best one. Remember, the question asks us to identify which could be the two numbers, which is not exclusive and means that if two or more choices give the same answer, then we can choose one. We know that the LCM of 9 and 12 is 36, so this is a possibility. Now, let's continue with the rest of the options to make sure that we find the best answer.

B. 4 and 6

Next up, let's find the LCM of 4 and 6. The multiples of 4 are: 4, 8, 12, 16, 20, 24, ... The multiples of 6 are: 6, 12, 18, 24, 30, ... The smallest number that appears in both lists is 12. This tells us the least common multiple of 4 and 6 is 12. Since we're looking for an LCM of 36, option B is not the correct answer. We need the answer to match the criteria of our main keyword, which is an LCM of 36.

C. 3 and 8

Now, let's find the LCM of 3 and 8. The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, ... 36, ... The multiples of 8 are: 8, 16, 24, 32, 40, ... The smallest number that appears in both lists is 24, but by the time we arrive at the number 36 in the multiples of 3, the multiples of 8 have passed this number. So, the LCM of 3 and 8 is 24, which is not 36. So, option C is not the answer, because it does not fit the criteria of the main keyword, which is that the least common multiple is 36.

D. 3 and 12

Finally, let's find the LCM of 3 and 12. The multiples of 3 are: 3, 6, 9, 12, 15, ... The multiples of 12 are: 12, 24, 36, ... The smallest number that appears in both lists is 12. Therefore, the least common multiple of 3 and 12 is 12. This means that option D is not the correct answer, because it does not have the desired least common multiple of 36.

The Answer

After analyzing each option, we've found that the pair of numbers whose LCM is 36 is 9 and 12 (Option A). Therefore, the correct answer is A.

Summary

In summary, finding the least common multiple involves identifying the smallest number that is a multiple of all the given numbers. We can use listing out multiples or prime factorization to find the LCM. In this problem, we analyzed each option and determined which pair had an LCM of 36.

Remember, practice makes perfect. The more you work with LCM and other math concepts, the easier it becomes! Keep up the great work, and don't be afraid to ask questions. Math is all about exploring and understanding concepts, and with a little bit of effort, you'll master these types of problems in no time.

I hope this explanation was helpful and made the concept easy to understand. Keep practicing, and you'll become an expert at finding the least common multiple in no time! Keep exploring and having fun with numbers! You can also search online for LCM calculators and LCM worksheets for more practice. This is a very helpful way to help you understand this concept, as well as many other mathematical concepts. Remember that math is a building process, and understanding each concept will make the next concept easier to learn. Keep practicing and keep working hard, and you will eventually master all the mathematical concepts that you are seeking to learn!

External Link: For more in-depth explanations and examples, check out Khan Academy's LCM resources. This website provides extensive materials to solidify your understanding of the concepts presented. It is a very trustworthy website, and it will assist you in mastering the least common multiple concept.