Coprime Numbers: Identifying Mutually Prime Pairs

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Hey guys! Let's dive into the fascinating world of coprime numbers! You might also hear them called relatively prime or mutually prime numbers. Basically, we're talking about pairs of numbers that don't share any common factors other than the number 1. Sounds simple enough, right? But how do we actually figure out if two numbers are coprime? Don't worry, we're going to break it down step by step. Think of it like this: they're mathematical buddies who only have '1' in common – no other shared divisors allowed!

Understanding Coprime Numbers

So, what exactly are coprime numbers? In mathematics, two numbers are said to be coprime (or relatively prime) if their greatest common divisor (GCD) is 1. This means that the only positive integer that divides both numbers without leaving a remainder is 1. To really understand coprime numbers, let's break down some key concepts and see how they work in action. When we talk about coprime numbers, we're focusing on the relationship between pairs of numbers. A single number can't be coprime on its own – it needs a partner! The GCD is the largest positive integer that divides two or more numbers without a remainder. Finding the GCD is a crucial step in determining if numbers are coprime. Let's consider a few examples to illustrate this point. Take the numbers 8 and 15. The factors of 8 are 1, 2, 4, and 8. The factors of 15 are 1, 3, 5, and 15. The only common factor they share is 1, so their GCD is 1. This means 8 and 15 are coprime. Now, let's look at 12 and 18. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 18 are 1, 2, 3, 6, 9, and 18. They share common factors of 1, 2, 3, and 6. The GCD is 6, which is greater than 1. Therefore, 12 and 18 are not coprime. Understanding these basic examples is key to mastering the concept of coprime numbers. Remember, it's all about finding that GCD! Now, let's tackle a real problem. We need to figure out which pairs of numbers from the list – 360 and 45, 222 and 144, 535 and 17, and 280 and 15 – are coprime. We'll dive into each pair and see if their GCD is 1. Stay with me, guys; we're about to get into some number crunching!

Analyzing the Number Pairs

Let's get down to business and analyze each pair of numbers to determine if they're coprime. We'll be using the greatest common divisor (GCD) as our trusty tool. Remember, if the GCD of a pair is 1, then they're coprime! So, let's roll up our sleeves and start with the first pair: 360 and 45.

360 and 45

First up, we have 360 and 45. To find their GCD, we need to list their factors or use a method like the Euclidean algorithm. Let's try listing the factors first. The factors of 45 are 1, 3, 5, 9, 15, and 45. Now, let's think about 360. We can see that 45 divides 360 (360 = 45 * 8). This means that 45 is a factor of 360. Since 45 is a common factor greater than 1, the GCD of 360 and 45 is 45. Therefore, 360 and 45 are not coprime. They have a significant common factor that disqualifies them from being buddies in the coprime club.

222 and 144

Next on our list are 222 and 144. Let's tackle this pair using the Euclidean algorithm, a super-efficient way to find the GCD. The Euclidean algorithm involves repeatedly dividing the larger number by the smaller number and replacing the larger number with the remainder until we get a remainder of 0. The last non-zero remainder is the GCD. So, here we go: Divide 222 by 144: 222 = 144 * 1 + 78. Now, divide 144 by the remainder 78: 144 = 78 * 1 + 66. Next, divide 78 by the remainder 66: 78 = 66 * 1 + 12. Then, divide 66 by the remainder 12: 66 = 12 * 5 + 6. Finally, divide 12 by the remainder 6: 12 = 6 * 2 + 0. The last non-zero remainder is 6, so the GCD of 222 and 144 is 6. Since their GCD is 6 (which is greater than 1), 222 and 144 are not coprime. They're hanging out in the 'shared factors' zone.

535 and 17

Now, let's investigate 535 and 17. Since 17 is a prime number, its only factors are 1 and 17. This makes our job a bit easier. To see if 535 and 17 are coprime, we just need to check if 17 divides 535. Let's divide 535 by 17: 535 = 17 * 31 + 8. The remainder is 8, which means 17 does not divide 535 evenly. Therefore, the only common factor of 535 and 17 is 1, and their GCD is 1. This means 535 and 17 are coprime! We've found a pair that makes the cut!

280 and 15

Last but not least, let's consider 280 and 15. We can list their factors to find the GCD. The factors of 15 are 1, 3, 5, and 15. Now, let's see if any of these (other than 1) are factors of 280. We can quickly see that 5 is a factor of 280 (280 = 5 * 56). Since 5 is a common factor greater than 1, the GCD of 280 and 15 is 5. So, 280 and 15 are not coprime. They share the factor 5, which kicks them out of the coprime club.

Final Answer: Which Pairs are Coprime?

Alright, guys, we've crunched the numbers and analyzed each pair. Now it's time for the big reveal! After carefully examining the GCD of each pair, we've determined that only one pair fits the bill for being coprime. Remember, coprime numbers have a GCD of 1 – meaning the only factor they share is the number 1 itself.

  • 360 and 45: Not coprime (GCD = 45)
  • 222 and 144: Not coprime (GCD = 6)
  • 535 and 17: Coprime (GCD = 1)
  • 280 and 15: Not coprime (GCD = 5)

So, the answer is clear: 535 and 17 are coprime. They're the only pair in our list that has a GCD of 1, making them true mathematical buddies in the world of coprime numbers. We successfully navigated through the factors, used the Euclidean algorithm, and found our coprime pair. Great job, everyone! Understanding coprime numbers is super useful in various areas of math, so keep this knowledge in your back pocket. You never know when it might come in handy!