Integer Values Of Expressions A Detailed Solution
Hey guys! Today, we're diving deep into a fascinating problem from the realm of mathematics. We're going to explore at how many integer values of n the expression ((n+5)(n-5)+31)/n yields integer values. This might sound intimidating at first, but don't worry, we'll break it down step by step, making it super easy to understand. Get ready to flex those mathematical muscles and learn something new!
Before we jump into solving the problem, let's make sure we understand the expression we're dealing with. We have ((n+5)(n-5)+31)/n. The key here is to simplify it as much as possible. Think of it as untangling a knot – the simpler the knot, the easier it is to undo! The first thing we can do is expand the (n+5)(n-5) part. Remember the difference of squares formula? It's like a magic trick that helps us simplify things quickly. This formula tells us that (a+b)(a-b) = a^2 - b^2. Applying this to our expression, we get n^2 - 25. So, now our expression looks like (n^2 - 25 + 31)/n. See? We're already making progress! Next, we can combine the constants, -25 and +31, which gives us +6. So, now our expression is (n^2 + 6)/n. This is much simpler, right? But we can simplify it even further! We can split the fraction into two parts: n^2/n + 6/n. And guess what? n^2/n simplifies to just n. So, our expression finally becomes n + 6/n. This is the simplified form we'll be working with. Remember, the goal is to find integer values of n for which this entire expression results in an integer. This means that both n and 6/n must be integers. The n + 6/n expression represents the core of our problem. We've transformed it from a seemingly complex form to a more manageable one. This simplification is a crucial step in solving many mathematical problems. By breaking down the expression, we've made it easier to analyze and identify the conditions that need to be met for it to yield integer values. The simplified form, n + 6/n, highlights the importance of the term 6/n. For the entire expression to be an integer, 6/n must also be an integer. This means that n must be a divisor of 6. In other words, n must be a number that divides 6 without leaving a remainder. This understanding is key to finding the possible values of n that satisfy the condition of the problem. Now that we have a simplified expression and a clearer understanding of the problem, we can move on to finding the integer values of n that make the expression an integer. This involves identifying the divisors of 6 and then checking each divisor to see if it satisfies the original condition. It's like detective work, where we follow the clues to find the solution! By simplifying the expression, we've not only made the problem easier to solve but also gained a deeper understanding of the relationship between n and the expression's value. This kind of simplification is a powerful tool in mathematics, allowing us to tackle complex problems with greater ease and confidence. Keep this in mind as we continue our journey to find the integer values of n!
Okay, now for the exciting part: finding the integer values of n. Remember, for n + 6/n to be an integer, 6/n must also be an integer. This means n has to be a divisor of 6. So, what are the divisors of 6? Well, they are the numbers that divide 6 perfectly, without leaving any remainder. Let's list them out, both positive and negative, because we're looking for integer values, and integers can be positive or negative. The divisors of 6 are: -6, -3, -2, -1, 1, 2, 3, and 6. That's a total of eight potential values for n. Each of these values, when divided into 6, results in an integer. For example, 6 divided by -6 is -1, 6 divided by -3 is -2, and so on. Similarly, 6 divided by 1 is 6, 6 divided by 2 is 3, and so forth. Now that we have these potential values, we need to check if they actually work in our simplified expression, n + 6/n. This is like testing our suspects to see if they fit the crime! Let's go through each value of n and see what we get:
- If n = -6, then
n + 6/n= -6 + (6/-6) = -6 - 1 = -7, which is an integer. - If n = -3, then
n + 6/n= -3 + (6/-3) = -3 - 2 = -5, which is an integer. - If n = -2, then
n + 6/n= -2 + (6/-2) = -2 - 3 = -5, which is an integer. - If n = -1, then
n + 6/n= -1 + (6/-1) = -1 - 6 = -7, which is an integer. - If n = 1, then
n + 6/n= 1 + (6/1) = 1 + 6 = 7, which is an integer. - If n = 2, then
n + 6/n= 2 + (6/2) = 2 + 3 = 5, which is an integer. - If n = 3, then
n + 6/n= 3 + (6/3) = 3 + 2 = 5, which is an integer. - If n = 6, then
n + 6/n= 6 + (6/6) = 6 + 1 = 7, which is an integer.
Guess what? All eight values work! Each one of them, when plugged into our expression, gives us an integer result. It's like we've cracked the code! This means there are eight integer values of n for which the expression ((n+5)(n-5)+31)/n yields an integer. Isn't that awesome? We started with a seemingly complex expression, simplified it, identified the key condition (that n must be a divisor of 6), and then systematically found all the values that satisfy the condition. This is the beauty of mathematics – breaking down problems into smaller, manageable steps and then solving them one by one. By listing out the divisors of 6 and then testing each one, we ensured that we didn't miss any possible solutions. This methodical approach is crucial in problem-solving, especially in mathematics. It helps us avoid making mistakes and ensures that we find all the correct answers. The process of checking each value also reinforces our understanding of the relationship between n and the expression's value. We can see how different values of n affect the result and why certain values lead to integer outcomes. This deeper understanding is invaluable, as it allows us to apply these concepts to other problems in the future. Now that we've found all eight integer values, we can confidently say that we've solved the problem. But the journey doesn't end here! The skills and techniques we've used in this problem can be applied to a wide range of mathematical challenges. So, keep practicing, keep exploring, and keep that mathematical curiosity alive!
So, guys, we've successfully navigated this mathematical puzzle! We discovered that there are eight integer values of n for which the expression ((n+5)(n-5)+31)/n yields integer values. That's a pretty neat accomplishment, right? We started by simplifying the expression, which was like clearing away the fog to see the path ahead. Then, we identified the crucial condition: n had to be a divisor of 6. This was like finding the key that unlocks the door to the solution. Finally, we systematically listed and checked all the divisors of 6, ensuring we didn't miss any possible answers. This was like carefully piecing together the puzzle to reveal the complete picture. This problem highlights the power of simplification in mathematics. By breaking down a complex expression into a simpler form, we made it much easier to analyze and solve. It's like taking a tangled ball of yarn and carefully untangling it strand by strand. Each step of simplification brings us closer to the solution. The process of identifying the divisors of 6 also reinforces the importance of understanding number theory concepts. Divisibility is a fundamental concept in mathematics, and it plays a crucial role in many problem-solving scenarios. By recognizing that n had to be a divisor of 6, we were able to narrow down the possibilities and focus our efforts on the most likely candidates. The methodical approach we used to check each divisor is also worth noting. We didn't just guess or assume; we systematically tested each value to ensure it satisfied the condition. This is a valuable skill in any problem-solving situation, whether it's in mathematics or in real life. It's about being thorough, careful, and not jumping to conclusions. We've not only solved a specific problem but also learned some valuable problem-solving techniques along the way. These techniques, such as simplification, identifying key conditions, and methodical checking, can be applied to a wide range of challenges. So, remember these lessons as you continue your mathematical journey. Keep practicing, keep exploring, and keep challenging yourself with new problems. The more you practice, the better you'll become at recognizing patterns, applying concepts, and finding creative solutions. And who knows, maybe you'll be the one teaching others how to solve these kinds of problems someday! So, let's celebrate our success in cracking this mathematical puzzle, and let's look forward to the next challenge with enthusiasm and confidence!
- Integer values
- Mathematical expressions
- Divisors
- Simplification
- Problem-solving