Prime Numbers: Find 'n' For (30 + 2n)/n

by Admin 40 views
Prime Numbers: Find 'n' for (30 + 2n)/n

Hey guys! Let's dive into a cool mathematical problem today. We're going to explore number theory and prime numbers. Our main goal is to determine all natural numbers, represented by 'n', that make the expression (30 + 2n) / n a prime number. This involves some clever algebraic manipulation and a solid understanding of what makes a number prime. So, grab your thinking caps, and let's get started!

Understanding the Problem

At its core, this problem asks us to find specific values of 'n' (which are natural numbers, meaning positive integers like 1, 2, 3, and so on) that satisfy a particular condition. That condition is that when you plug 'n' into the expression (30 + 2n) / n, the result must be a prime number. Remember, a prime number is a whole number greater than 1 that has only two divisors: 1 and itself. Examples of prime numbers include 2, 3, 5, 7, 11, and so on.

To really nail this, we need to break down the expression and see how different values of 'n' affect the outcome. The expression (30 + 2n) / n looks a bit intimidating at first, but we can simplify it. This simplification is key to figuring out which 'n' values will give us prime numbers. We'll use a little bit of algebra to make it easier to work with. So, let's roll up our sleeves and get into the nitty-gritty of the math!

Simplifying the Expression

Okay, so we have the expression (30 + 2n) / n. The trick here is to split the fraction into two separate fractions. This is a classic algebraic technique that helps us see the structure more clearly. We can rewrite the expression like this:

(30 + 2n) / n = 30 / n + 2n / n

Now, this looks a bit better! We've separated the expression into two terms. The next step is to simplify each term individually. The first term, 30 / n, is straightforward. It simply represents 30 divided by 'n'. The second term, 2n / n, is even easier. Notice that we have 'n' in both the numerator and the denominator. This means we can cancel them out:

2n / n = 2

So, our simplified expression becomes:

30 / n + 2

This simplified form is much easier to work with. It tells us that the value of our original expression depends on the value of 30 / n. Specifically, we need to find values of 'n' that, when divided into 30, give us a result that, when added to 2, yields a prime number. This gives us a clear path forward.

Identifying Potential Values of 'n'

Now that we have the simplified expression 30 / n + 2, we can start thinking about what values of 'n' would make this whole thing a prime number. Remember, 'n' has to be a natural number (1, 2, 3, ...), and we need the entire expression to result in a prime number. This means 30 / n must be a whole number, as we're adding it to 2, which is already a whole number. So, 'n' must be a divisor (or factor) of 30. Let's list out the divisors of 30:

1, 2, 3, 5, 6, 10, 15, 30

These are all the natural numbers that divide 30 without leaving a remainder. Now, we need to test each of these values of 'n' in our simplified expression, 30 / n + 2, to see if the result is a prime number. This is where the fun begins! We'll systematically go through each divisor and check if it fits the bill. This process of elimination will lead us to the correct solutions.

Testing the Divisors of 30

Alright, let's put those divisors of 30 to the test! We'll go through each one, plug it into our simplified expression (30 / n + 2), and see if we get a prime number. This is a crucial step in solving the problem, so let's take our time and be meticulous.

  • n = 1: 30 / 1 + 2 = 30 + 2 = 32 (Not prime)
  • n = 2: 30 / 2 + 2 = 15 + 2 = 17 (Prime!)
  • n = 3: 30 / 3 + 2 = 10 + 2 = 12 (Not prime)
  • n = 5: 30 / 5 + 2 = 6 + 2 = 8 (Not prime)
  • n = 6: 30 / 6 + 2 = 5 + 2 = 7 (Prime!)
  • n = 10: 30 / 10 + 2 = 3 + 2 = 5 (Prime!)
  • n = 15: 30 / 15 + 2 = 2 + 2 = 4 (Not prime)
  • n = 30: 30 / 30 + 2 = 1 + 2 = 3 (Prime!)

Wow, we've found some prime numbers! It looks like n = 2, 6, 10, and 30 all result in prime numbers when plugged into our expression. This is a significant breakthrough. We've narrowed down the possible values of 'n' that satisfy our condition. Now, let's make sure we haven't missed anything and then present our final answer.

Final Answer

Okay, guys, after carefully testing all the divisors of 30, we've found the natural numbers 'n' that make the expression (30 + 2n) / n a prime number. Let's recap our findings:

  • When n = 2, (30 + 2n) / n = 17 (Prime)
  • When n = 6, (30 + 2n) / n = 7 (Prime)
  • When n = 10, (30 + 2n) / n = 5 (Prime)
  • When n = 30, (30 + 2n) / n = 3 (Prime)

Therefore, the natural values of 'n' for which the expression (30 + 2n) / n is a prime number are 2, 6, 10, and 30. We've successfully solved the problem! This journey involved simplifying the expression, identifying potential values of 'n', and then meticulously testing each one. It's a testament to the power of combining algebra and number theory. Great job, everyone, for sticking with it! You've conquered a challenging problem today.