We have been studying proof by contradictions and one of the practice problems is to prove the above proposition using proof by contradiction. The intial line I have is:
We will use proof by contradiction. We assume the proposition is false, which means n is an integer greater than 2, and there exist an integer m such that n divides m and n + m = nm.
Any suggestions on what the next step should be to reach a contradiction?
$n+m=mn$ implies that $1/n+1/m=1$. This means that when $n>2$, $$ \frac{1}{m}=1-\frac{1}{n}>1-\frac{1}{2}=\frac{1}{2},m<2. $$ We also have $1/m<1/n+1/m=1,m>1$. So $m$ is NOT an integer, a contradiction.
Your "n does not divide m" part is confusing since everything divides 1. Note that you also have some confusion between "and" and "or". The sentence "which means n is an integer greater than 2, and there exist an integer m such that n divides m and n + m = nm" should be "which means n is an integer greater than 2, and there exist an integer m such that n divides m OR n + m = nm".