I was looking at a past paper and I stumbled upon this question, and I was very confused.
Here’s the question:
A number is the “reciprocal” of another number if their product is equal to 1. Suppose the sum of a reciprocal of a positive integer $m$ and the reciprocal of 2 is equal to the sum of the reciprocals of two integers that are different from $m$. How many possible $m$ would satisfy this condition?
I made this formula, I’m not sure if this is the correct way to do it ($n$ and $p$ are the two other positive integers besides $m$, also unknown) $$\frac{1}{m}+\frac{1}{2}=\frac{1}{n}+\frac{1}{p},\ m > 0,\ n \neq m,\ p \neq m $$
I’ve tried isolating $m$, but I don’t know how to continue from there. $$m=\frac{1}{\frac{1}{n}+\frac{1}{p}-\frac{1}{2}}$$
In addition, I don’t really feel like this is the right approach, since this equation can’t really tell me how many different results there are going to be. I would really appreciate some help!
EDIT: This is supposed to be a test question, so methods like running through each possible possibility is not really plausible.