Common Multiples: Finding The Multiple Of 9 And 14

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Hey guys! Let's dive into a cool math problem today that involves finding common multiples. This is super useful in many real-life situations, from scheduling events to figuring out how many items you need to buy to have equal amounts of different things. We're going to break down how to find a common multiple of 9 and 14, and then pick the correct answer from the given options. So, grab your thinking caps, and let's get started!

Understanding Multiples and Common Multiples

First, let's define what multiples are. A multiple of a number is simply the result you get when you multiply that number by an integer (a whole number). For example, the multiples of 9 are 9, 18, 27, 36, and so on. Similarly, the multiples of 14 are 14, 28, 42, 56, and so on. A common multiple is a number that is a multiple of two or more numbers. For instance, if we list out the multiples of 9 and 14, any number that appears in both lists is a common multiple.

To effectively solve problems related to multiples, it's essential to understand the underlying concepts. When dealing with multiples, we're essentially looking at numbers that can be obtained by multiplying a given number by an integer. This can be quite helpful in various real-world scenarios, such as when you're trying to determine how many items you need to purchase to ensure you have the same quantity of different things. Think about it this way: if you're planning a party and need to buy both plates and napkins, finding a common multiple can help you figure out the minimum number of packs you need to buy to have an equal amount of each.

Moreover, the concept of multiples is closely tied to divisibility. If a number is a multiple of another number, it means it is divisible by that number without leaving any remainder. This connection between multiples and divisibility makes it easier to identify common multiples, as you can quickly check whether a number is divisible by all the numbers in question. This understanding can significantly simplify problem-solving and make mathematical concepts more intuitive.

Methods to Find Common Multiples

There are a couple of ways we can find common multiples. One method is to list the multiples of each number until we find a common one. This works well for smaller numbers, but it can become tedious for larger numbers. Another, more efficient method involves finding the Least Common Multiple (LCM). The Least Common Multiple (LCM) is the smallest number that is a multiple of both numbers. Once we find the LCM, we can easily find other common multiples by multiplying the LCM by any integer. Let's explore both methods to give you a comprehensive understanding.

Listing Multiples

To find common multiples by listing, you simply write out the multiples of each number until you spot a match. This approach is straightforward and easy to understand, making it a great starting point for grasping the concept of common multiples. For smaller numbers, this method is quite efficient. However, as the numbers get larger, listing multiples can become time-consuming and prone to errors. You might end up writing out dozens of multiples before you find a common one, which can be quite tedious. For example, if you were trying to find the common multiples of 24 and 36, you'd need to list quite a few multiples of each before you'd find the smallest common multiple, which is 72. Despite its limitations with larger numbers, listing multiples is an excellent way to visually understand what common multiples are and how they relate to the original numbers.

Finding the Least Common Multiple (LCM)

Finding the Least Common Multiple (LCM) is a more systematic approach, especially when dealing with larger numbers. The LCM is the smallest number that is a multiple of both given numbers. Once you find the LCM, you can easily determine other common multiples by multiplying the LCM by any whole number. There are several methods to find the LCM, but one of the most common is the prime factorization method. This involves breaking down each number into its prime factors and then using those factors to construct the LCM.

To illustrate, let's consider the numbers 12 and 18. The prime factorization of 12 is 2 × 2 × 3, and the prime factorization of 18 is 2 × 3 × 3. To find the LCM, you take the highest power of each prime factor that appears in either factorization. In this case, the highest power of 2 is 2², the highest power of 3 is 3². So, the LCM is 2² × 3² = 4 × 9 = 36. Once you have the LCM, finding other common multiples is easy. For instance, the next common multiple would be 36 × 2 = 72, and so on. Using the LCM method not only simplifies the process but also ensures you find the smallest common multiple, which can be particularly useful in various mathematical problems and real-life applications.

Solving the Problem

Now, let's apply these methods to our problem: finding a common multiple of 9 and 14 from the options A) 144, B) 126, C) 84, and D) 52.

Method 1: Listing Multiples (Partial)

We can start by listing multiples of 9 and 14 to see if any of the options appear in both lists. This can quickly help us narrow down the possibilities.

Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126... Multiples of 14: 14, 28, 42, 56, 70, 84, 98, 112, 126...

We see that 126 appears in both lists!

Method 2: Finding the LCM

Let's also find the LCM of 9 and 14 to confirm our answer. This will give us a more robust understanding and check if 126 is indeed a common multiple.

First, find the prime factorization of each number:

9 = 3 x 3 = 3² 14 = 2 x 7

Now, take the highest power of each prime factor:

2¹ (from 14) 3² (from 9) 7¹ (from 14)

Multiply these together to get the LCM:

LCM (9, 14) = 2 x 3² x 7 = 2 x 9 x 7 = 126

The LCM of 9 and 14 is 126. This means 126 is the smallest common multiple of 9 and 14. Since 126 is an option, it is indeed a common multiple.

Analyzing the Options

Now that we've found a common multiple, let's go through the options to make sure we've got the right one and understand why the others might not fit.

  • A) 144: 144 is a multiple of 9 (144 = 9 x 16), but it's not a multiple of 14. If you divide 144 by 14, you get approximately 10.29, which is not a whole number. So, 144 can't be a common multiple.
  • B) 126: We've already shown that 126 is a multiple of both 9 and 14. It's in both lists of multiples, and it's the LCM, so it's definitely a common multiple.
  • C) 84: 84 is a multiple of 14 (84 = 14 x 6), but it's not a multiple of 9. When you divide 84 by 9, you get approximately 9.33, which isn't a whole number. Thus, 84 is not a common multiple.
  • D) 52: 52 is not a multiple of either 9 or 14. If you divide 52 by 9, you get approximately 5.78, and if you divide 52 by 14, you get approximately 3.71. Neither result is a whole number, so 52 is not a common multiple.

Final Answer

From our calculations and analysis, we can confidently say that the correct answer is B) 126. This is the only number in the options that is a multiple of both 9 and 14. Understanding how to find common multiples is super useful, not just in math class, but in everyday life too!

So, there you have it! We've walked through the steps to find a common multiple of 9 and 14 and identified the correct answer. Keep practicing these methods, and you'll become a pro at finding common multiples in no time. Great job, guys! Keep up the awesome work! Understanding these concepts is crucial, and you're doing fantastic!