I need to show that Given $n>0$ and a rational number $q$ there are only finitely many n-tuples $(c_1,...,c_n)$ of natural numbers such that $q=1/c_1+...+1/c_n$. This result can be used to show that there are finitely many groups with a fixed number of conjugacy classes. i.e, we have a group $G$ of size $lcm(c_1,...,c_n)$ for each n-tuple$(c_1,...,c_n)$ such that $1=\frac{1}{c_1}+\frac{1}{c_2}+...+\frac{1}{c_n}$.
Ok so for example, when $n=5$ and $q=1$, $1=\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}$, so $(5,5,5,5,5)$ is a 5-tuple that work. There might be other.We need to show there are finitely many of them.
It is in remark in the following picture:
The smallest $c_i$ must be at most $\frac{n}{q}$ -- otherwise the $\frac1{c_i}$ are all so small that the sum of them cannot reach $q$. Therefore there are only finitely many choices for what the smallest $c_i$ can be. For each of those choices, use induction on $n$ to conclude that there are only finitely many ways to fill out the rest of the $c_j$s.