I am trying to get some practice on disproving existence statements and I was really stuck on this one:
"There exists an example of three distinct positive integers different from $a,2a,3a$ for some $a \in \mathbb{N}$ having the property that each divides the sum of the other two."
I have tried working with the negation for all distinct positive integers different from $a,2a,3a$ for some $a \in \mathbb{N}$ where it each does not divide the sum of the other two, and thought about it using the contrapositive, except that I am not quite sure how to start it.
In fact I was rather stumped on expressing it in generalized terms. Would someone mind helping me out with this ? I think this is quite a good example and was quite difficult for me so I would really like to know the best way to approach such a problem.
Help is greatly appreciated.
One should not worry too much about logic (negation, contrapositives), and instead think about the concrete problem, that is, just think about numbers. What can we discover about numbers such the sum of any two is divisible by the third?
If $x,y,z$ are positive integers, with $x\lt y\lt z$, then it looks "hardest" for $z$ to divide $x+y$. Since $x+y\lt 2z$, the only way we can have divisibility is if $x+y=z$. Nice simplification!
Now we want $y$ to divide $x+z$, so we want $y$ to divide $2x+y$. That can only happen if $y$ divides $2x$. But since $x\lt y$, that forces $2x=y$, and we are finished.