Finding The Natural Number 'M': A Math Problem Solved!

Hey guys! Let's dive into a fun math problem. We're given a natural number, which we'll call 'M'. We know that there are three different natural numbers that are multiples of 'M' and are also less than 200. These multiples are 45, 105, and 150. The question is: what is the value of 'M'? This problem tests our understanding of factors, multiples, and how they relate to each other. So, let's break it down and find the solution! This is not just about finding an answer; it's about understanding the relationships between numbers. The cool thing about math is that once you understand the core concepts, solving problems like this becomes much easier. We'll go through the steps logically so you can follow along and apply the same method to similar problems in the future. Remember, practice makes perfect, and the more you work through problems, the more confident you'll become in your math skills. So, let's get started and unravel this mathematical mystery! The key to solving this problem lies in understanding the concept of factors. A factor is a number that divides another number completely, without leaving any remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. These are all the numbers that can divide 12 evenly. In our problem, 'M' must be a factor of 45, 105, and 150. This is because 45, 105, and 150 are multiples of 'M'. So, we are essentially looking for a common factor of these three numbers. Understanding this relationship is crucial for solving the problem correctly. We will explore how to identify the greatest common factor (GCF), which will lead us to the solution. This method allows us to find the largest possible value for 'M', ensuring that it satisfies all the given conditions. Let's start by listing the factors of each number to identify the common ones.

Unveiling the Factors: The First Step

Alright, let's get our hands dirty and list the factors of each of the numbers provided: 45, 105, and 150. This is a crucial step because it helps us visualize the numbers that divide each of these evenly. When we list the factors, we want to make sure we find all of them to make the right connection to the value 'M'. Remember, 'M' is a factor of all three numbers, so it must be a number that divides all of them without leaving any remainder. This process allows us to systematically find the common factors, leading us closer to the correct answer. Let's start with 45. The factors of 45 are: 1, 3, 5, 9, 15, and 45. Now, let's move on to 105. The factors of 105 are: 1, 3, 5, 7, 15, 21, 35, and 105. Finally, let's consider 150. The factors of 150 are: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, and 150. Looking at these lists, we're looking for common numbers among all three lists. We can see that the common factors of 45, 105, and 150 are 1, 3, 5, and 15. Since the problem tells us that 45, 105, and 150 are multiples of M and asks us to find M, we need to choose one of these common factors for M. However, knowing that 45, 105, and 150 are multiples of M, then M must divide them. The value of 'M' must be a common factor of all three numbers. This part of the problem involves the basic principles of divisibility, so it's really important to get this step correct. The next step is to use the common factors to understand the value of M.

Identifying the Greatest Common Factor (GCF)

From the factors we found, we can see that the common factors of 45, 105, and 150 are 1, 3, 5, and 15. Since the question asks for the value of M, and we know that M must be a common factor, we must select from these options. However, the problem specifies that there are three different multiples of M that are less than 200. Let's examine this carefully. If M = 1, then the multiples would be 45, 105, and 150 which satisfy the condition. If M = 3, then the multiples are 45, 105, and 150. If M = 5, then the multiples are 45, 105, and 150. If M = 15, then the multiples are 45, 105, and 150. The numbers 45, 105, and 150 are multiples of all these common factors, so any of these numbers would fit the criteria and be correct. So, the question is which number is the best fit? The value of M is not limited to the GCF in this case because each of these fits the requirements of the problem. However, to solve the problem, we need to ensure that there are three different multiples of M less than 200. This is the crucial part. We know the multiples of M are 45, 105, and 150. If M = 1, the multiples are 45, 105, and 150. These are all different and less than 200. If M = 3, then the multiples are 45, 105, and 150, which are also different and less than 200. The same goes for M = 5 and M = 15. The problem does not have a unique solution given the information. Therefore, since M can be any of the common factors, but none of the information can determine the value of 'M', the answer is the lowest factor available. The value of M is 15.

Solving for 'M': The Final Answer

Based on our analysis, we know that M must be a factor of 45, 105, and 150. We found that the common factors are 1, 3, 5, and 15. Looking at the multiples of each of these values, we determine that all common factors fit within the problem, so the value of 'M' is 15. Since the question specifies three different multiples, and all the common factors have three or more multiples under 200, any of the factors will work for the value of M, and 15 is the highest value. Therefore, the answer is: M = 15. So there you have it, folks! We've successfully solved the problem by systematically identifying the factors, finding the common ones, and ensuring that the multiples fit the given conditions. This wasn't too hard, right? Remember, the key to solving such problems is to break them down into smaller, manageable steps. Always start by understanding what the problem is asking, then use the information provided to identify the relevant concepts. In this case, it was factors, multiples, and common factors. The problem could be about factors, multiples, and finding the greatest common factor. Once you have a firm grasp of these concepts, you'll be well-equipped to tackle similar problems with confidence. Keep practicing, and you'll become a math whiz in no time! Keep going and you'll be the math master!