Discovering Integer Solutions to Equations: Prime or Not?

In summary, the question is asking for distinct integers m and n such that 1/m + 1/n is an integer. By letting m = n = 1, we can see that this condition is satisfied. However, there is a more difficult question that asks for an integer n such that 2n^2-5n+2 is prime. To solve this, we need to understand the definition of a prime number and use the factorization (n-2)(2n-1) to find a suitable value for n. By letting n = 3, we can see that the expression evaluates to a prime number, thus proving the statement. It is important to note the use of distinct integers and not just any two
  • #1
mr_coffee
1,629
1
Hello everyone.

I'm suppose to prove this but I'm having troubles figuring out how u find "distinct" integers. Meaning they can't be the same number. i figured it out they just wanted integers though. Here is the question:
There are distinct integers m and n such that 1/m + 1/n is an integer.

I wrote:
Let m = n = 1. Then m and n are integers such that 1/m + 1/n = 1/1 + 1/1 = 2, which is an integer.

Is there a processes to figuring these things out or is it a guessing game?

Also a harder one is this one:
There is an integer n such that 2n^2-5n+2 is prime.

I looked up what the defintion of a prime number is and i got the following:
An integer n is prime if and only if n > 1 and for all positive integers r and s, if n = (r)(s), then r = 1, or s = 1.

So i wasn't sure where to start with that so I tried to factor 2n^2+5n+2 to see what happens and i got: (x-2)(2x-1). x = 2 or x = 1/2. Because 1/2 is not greater than 1 (x = 1/2) does this mean the whole expression is also not prime? Is that enough to prove it? IT says there IS an integer n that makes that expression prime though.
 
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  • #2
-1 is an integer. And remember, you're not trying to find a root of 2n^2-5n+2, you're looking for an n such that this evaluates to a prime number. Your factorization will help: try to make one of the factors 1 and the other a prime number.
 
  • #3
Do you have trouble with the word "distinct"? 1 and 1 are not distinct!
 
  • #4
Thanks for the help guys!
By letting m = 1, and n = -1, u get 0 which is an integer. For the 2nd one, am I allowed to just let as you suggested, (n-2)(2n-1); n-2 = 1, 2n-1 = 3;
n = 3, or n = 2. If you plug in 3 for n, u get 2(3)^2-5(3)+2 = 5, which is prime. So by example this is true?
 

1. What are integer solutions to equations?

Integer solutions to equations are the values of variables that satisfy the equation and result in a whole number. For example, in the equation 2x + 4 = 10, the integer solution would be x = 3.

2. How do I determine if an equation has integer solutions?

To determine if an equation has integer solutions, you can solve the equation and see if the resulting values for the variables are whole numbers. If they are, then the equation has integer solutions.

3. What is the significance of prime numbers in integer solutions?

Prime numbers play a significant role in integer solutions because they are numbers that are only divisible by 1 and themselves. This means that when trying to find integer solutions to an equation, prime numbers are the most basic building blocks that can be used to create other solutions.

4. Can an equation have both prime and non-prime integer solutions?

Yes, an equation can have both prime and non-prime integer solutions. This is because prime numbers are not the only numbers that can be used to create integer solutions. Non-prime numbers can also be used in combinations with other numbers to create solutions.

5. How can I use prime numbers to solve equations with integer solutions?

To use prime numbers to solve equations with integer solutions, you can start by factoring the equation and looking for common factors that are prime numbers. Then, you can use these prime numbers to create different combinations that satisfy the equation and result in integer solutions.

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