Natural Numbers Relatively Prime To 15 Between 200 And 300

Hey guys! Ever stumbled upon a math problem that seems like a riddle wrapped in an enigma? Well, today we're diving into one of those intriguing questions: "How many natural numbers from 200 to 300 are relatively prime to the number 15?" Sounds like a mouthful, right? But don't worry, we're going to break it down step by step, making it as clear as a sunny day. So, grab your thinking caps, and let's get started!

Understanding the Basics: Relatively Prime Numbers

Before we jump into solving the problem, let's make sure we're all on the same page about what it means for two numbers to be relatively prime, also known as coprime. Two numbers are said to be relatively prime if their greatest common divisor (GCD) is 1. In simpler terms, they don't share any common factors other than 1. For example, 8 and 15 are relatively prime because their only common factor is 1. However, 12 and 18 are not relatively prime because they share common factors like 2, 3, and 6. Understanding this concept is crucial because it forms the foundation for solving our main problem.

Now, let's think about the number 15. What are its prime factors? Well, 15 can be broken down into 3 and 5 (15 = 3 × 5). This is super important because any number that shares a factor of 3 or 5 with 15 will not be relatively prime to it. Therefore, to find the numbers that are relatively prime to 15, we need to identify and exclude the numbers between 200 and 300 that are divisible by either 3 or 5. This might sound a bit complex, but we'll tackle it methodically. We'll start by figuring out how many numbers in our range are divisible by 3, then how many are divisible by 5, and finally, how many are divisible by both. This will allow us to use the principle of inclusion-exclusion to find our answer. So, stay with me, and let's unravel this puzzle together!

Identifying Numbers Divisible by 3 and 5

Okay, let's get down to the nitty-gritty of finding the numbers between 200 and 300 that are divisible by 3 and 5. This is a crucial step in solving our problem, so let's take it slow and make sure we understand each part.

First, let's tackle the numbers divisible by 3. To find the first number in our range (200 to 300) that's divisible by 3, we can divide 200 by 3. The result is approximately 66.67. Since we need a whole number, we round up to the next integer, which is 67. So, the first multiple of 3 in our range is 67 × 3 = 201. Now, to find the last number in our range that's divisible by 3, we divide 300 by 3, which gives us 100. So, the last multiple of 3 in our range is 100 × 3 = 300. Now we know that the multiples of 3 in our range are 201, 204, 207, and so on, up to 300. To find out exactly how many numbers there are, we can use a simple formula: (Last multiple - First multiple) / 3 + 1. Plugging in our values, we get (300 - 201) / 3 + 1 = 99 / 3 + 1 = 33 + 1 = 34. So, there are 34 numbers between 200 and 300 that are divisible by 3.

Next, let's find the numbers divisible by 5. We follow a similar process. Divide 200 by 5, which gives us 40. So, the first multiple of 5 in our range is 40 × 5 = 200. Divide 300 by 5, which gives us 60. So, the last multiple of 5 in our range is 60 × 5 = 300. The multiples of 5 in our range are 200, 205, 210, and so on, up to 300. Using our formula, we get (300 - 200) / 5 + 1 = 100 / 5 + 1 = 20 + 1 = 21. So, there are 21 numbers between 200 and 300 that are divisible by 5.

But wait, we're not done yet! We also need to consider the numbers that are divisible by both 3 and 5. Why? Because we don't want to count them twice. These numbers are multiples of the least common multiple (LCM) of 3 and 5, which is 15. So, in the next section, we'll figure out how many numbers are divisible by 15, and then we'll use all this information to solve our original problem. Stick with me, we're getting closer!

Accounting for Numbers Divisible by Both 3 and 5 (Multiples of 15)

Alright, let's tackle the crucial step of figuring out how many numbers between 200 and 300 are divisible by both 3 and 5. As we discussed earlier, these numbers are multiples of the least common multiple (LCM) of 3 and 5, which is 15. We need to identify these numbers so we don't double-count them when we're calculating the total number of integers relatively prime to 15.

To find the first multiple of 15 within our range, we divide 200 by 15. This gives us approximately 13.33. Rounding up to the nearest whole number, we get 14. So, the first multiple of 15 in our range is 14 × 15 = 210. To find the last multiple of 15, we divide 300 by 15, which gives us 20. Therefore, the last multiple of 15 in our range is 20 × 15 = 300. The multiples of 15 in our range are 210, 225, 240, and so on, up to 300. To determine the total count of these multiples, we use our trusty formula: (Last multiple - First multiple) / 15 + 1. Plugging in the values, we get (300 - 210) / 15 + 1 = 90 / 15 + 1 = 6 + 1 = 7. So, there are 7 numbers between 200 and 300 that are divisible by 15.

Now that we've identified the numbers divisible by 3, 5, and 15, we have all the pieces of the puzzle. We're ready to use the principle of inclusion-exclusion to find the number of integers in our range that are not relatively prime to 15. This will then allow us to calculate the number of integers that are relatively prime to 15. We're in the home stretch, guys! Let's move on to the final calculations.

The Principle of Inclusion-Exclusion: Putting It All Together

Okay, guys, this is where all our hard work comes together! We're going to use the principle of inclusion-exclusion to figure out how many numbers between 200 and 300 are not relatively prime to 15. Remember, this principle helps us count things accurately when we have overlapping sets. In our case, the sets are numbers divisible by 3 and numbers divisible by 5.

The principle of inclusion-exclusion states that to find the total number of elements in the union of two sets, we add the number of elements in each set and then subtract the number of elements in their intersection. In our context:

Total numbers not relatively prime to 15 = (Numbers divisible by 3) + (Numbers divisible by 5) - (Numbers divisible by both 3 and 5)

We've already calculated these values:

• Numbers divisible by 3: 34
• Numbers divisible by 5: 21
• Numbers divisible by 15 (both 3 and 5): 7

Plugging these into our formula, we get:

Total numbers not relatively prime to 15 = 34 + 21 - 7 = 48

So, there are 48 numbers between 200 and 300 that are not relatively prime to 15. But remember, our original question asked for the number of integers that are relatively prime to 15. To find this, we need to subtract the number of non-relatively prime numbers from the total number of integers in our range.

How many integers are there between 200 and 300, inclusive? That's simply 300 - 200 + 1 = 101. Now we can calculate the number of integers relatively prime to 15:

Numbers relatively prime to 15 = (Total numbers in range) - (Numbers not relatively prime to 15)

Numbers relatively prime to 15 = 101 - 48 = 53

The Grand Finale: Finding the Solution

Drumroll, please! After all our calculations, we've finally arrived at the answer. We've determined that there are 53 natural numbers between 200 and 300 that are relatively prime to 15. That's quite a journey we've been on, from understanding the basics of relatively prime numbers to applying the principle of inclusion-exclusion!

So, if you were faced with this question on a test or in a math competition, you'd confidently select option A) 53. You've not only found the answer but also gained a solid understanding of the underlying concepts and techniques.

Remember, the key to solving complex math problems is to break them down into smaller, manageable steps. We started by defining relatively prime numbers, then identified the numbers divisible by 3 and 5, accounted for multiples of 15, and finally used the principle of inclusion-exclusion to reach our solution. By following this systematic approach, you can tackle even the most daunting mathematical challenges.

I hope this comprehensive guide has been helpful and has made the concept of relatively prime numbers a little less mysterious. Keep practicing, keep exploring, and most importantly, keep enjoying the fascinating world of mathematics! Until next time, mathletes!