Common Multiples Of 12 & 18 Under 150: Find The Count!

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Common Multiples of 12 & 18 Under 150: Find the Count!

Hey guys! Ever wondered how many numbers two different numbers can both divide into evenly, especially when there's a limit? Today, we're diving into a fun math problem: figuring out how many common multiples 12 and 18 share, but only looking at those smaller than 150. Let's break it down step by step so it's super easy to understand. We'll use some basic math principles and a bit of logical thinking to solve this. So, grab your thinking caps, and let's get started!

Understanding Multiples

Alright, so what exactly are multiples? Multiples are simply what you get when you multiply a number by an integer (that's a whole number, positive or negative, including zero). Think of it like this: the multiples of 12 are 12 x 1, 12 x 2, 12 x 3, and so on. That gives us 12, 24, 36, and so on. For 18, the multiples are 18 x 1, 18 x 2, 18 x 3, giving us 18, 36, 54, and so on. To find the common multiples of 12 and 18, we need to identify the numbers that appear in both lists. This is where the concept of the Least Common Multiple (LCM) comes in handy. The LCM is the smallest number that is a multiple of both 12 and 18. Finding the LCM will help us identify all the other common multiples more easily. Remember, each multiple is found by incrementally multiplying a number. Understanding this foundational concept is crucial for tackling the problem at hand. In more complex mathematical scenarios, especially when dealing with fractions or algebraic equations, knowing how to find multiples helps simplify the problems and identify patterns. This fundamental concept is not just useful for solving textbook questions but also in real-world applications, such as scheduling, resource allocation, and even in understanding musical harmony.

Finding the Least Common Multiple (LCM)

To efficiently pinpoint the common multiples, let's first nail down the Least Common Multiple (LCM) of 12 and 18. There are a couple of ways to do this, but one of the easiest is to list the multiples of each number until we find one they share. We already started doing this above! Let's write out a few more multiples for each:

  • Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, ...
  • Multiples of 18: 18, 36, 54, 72, 90, 108, 126, 144, ...

Notice that 36 appears in both lists. So, 36 is the LCM of 12 and 18! Another way to find the LCM is by using prime factorization. First, we break down 12 and 18 into their prime factors:

  • 12 = 2 x 2 x 3 = 2² x 3
  • 18 = 2 x 3 x 3 = 2 x 3²

Then, the LCM is found by taking the highest power of each prime factor that appears in either factorization: LCM = 2² x 3² = 4 x 9 = 36. Both methods confirm that the LCM of 12 and 18 is 36. Knowing the LCM is super useful, because every other common multiple of 12 and 18 will be a multiple of 36. This simplifies our task significantly. The LCM serves as a building block; it's the foundation upon which all other common multiples are built. By identifying this smallest shared multiple, we create a pathway to find larger ones efficiently. Understanding prime factorization and its application in finding the LCM is a powerful tool in number theory and can be applied in various other mathematical contexts.

Identifying Common Multiples Less Than 150

Now that we know the LCM of 12 and 18 is 36, finding all the common multiples less than 150 becomes much easier. Remember, any multiple of the LCM will also be a common multiple of the original numbers. So, we need to find all the multiples of 36 that are less than 150. Let’s list them out:

  • 36 x 1 = 36
  • 36 x 2 = 72
  • 36 x 3 = 108
  • 36 x 4 = 144
  • 36 x 5 = 180

Notice that 180 is greater than 150, so we stop at 144. Therefore, the common multiples of 12 and 18 that are less than 150 are 36, 72, 108, and 144. How many are there? Just count them: 1, 2, 3, 4. There are four common multiples. It's important to verify each number to ensure accuracy. By ensuring that each multiple is indeed a multiple of both 12 and 18 and that it falls within the specified limit of 150, we can confidently present the final count. Understanding this iterative process is a fundamental skill that can be transferred to more complex mathematical challenges. In various fields of study, such as engineering and finance, similar iterative processes are used to optimize solutions, manage constraints, and make informed decisions. Moreover, these foundational skills are not just confined to academic exercises; they enable problem-solving capabilities that extend to real-world scenarios.

The Answer

So, after breaking it all down, we found that there are 4 common multiples of 12 and 18 that are less than 150. These multiples are 36, 72, 108, and 144. Wasn't that a fun little math adventure? By understanding multiples, finding the LCM, and systematically listing out the common multiples, we solved the problem!

Conclusion

And there you have it! Finding the common multiples of two numbers within a certain range is a straightforward process once you understand the basics. It involves identifying multiples, finding the LCM, and then listing all the multiples of the LCM that fall within the specified limit. This problem-solving approach is valuable not only in mathematics but also in various real-life situations where identifying common patterns or occurrences is necessary. Whether you are scheduling events, managing resources, or analyzing data, the ability to identify common multiples can help you make informed decisions and optimize outcomes. In the grand scheme of things, grasping the essence of multiples and LCMs provides a solid foundation for tackling more advanced mathematical concepts. So, keep practicing and exploring the fascinating world of numbers! Remember, every mathematical problem is a stepping stone to enhance your problem-solving abilities. Embracing this mindset will make learning math not just easier but also more enjoyable. Keep exploring, keep questioning, and keep expanding your mathematical horizons!