Proper isometry group/Finite subgroup/Numerical properties/Possibilities/Fact

The numerial equation

${}2{\left(1-{\frac {1}{n}}\right)}=\sum _{i=1}^{m}{\left(1-{\frac {1}{n_{i}}}\right)}\,$

with

${}n\geq 2,\,m\in \mathbb {N}$ and with ${}2\leq n_{1}\leq n_{2}\leq \ldots \leq n_{m}\leq n$ has the following integer solutions.

1. ${}m=2$, and ${}n=n_{1}=n_{2}$.
2. In case ${}m=3$, there are the possibilities
   1. ${}n_{1}=n_{2}=2$, and ${}n=2n_{3}$,
   2. ${}n_{1}=2$, ${}n_{2}=n_{3}=3$, and ${}n=12$,
   3. ${}n_{1}=2$, ${}n_{2}=3$, ${}n_{3}=4$, and ${}n=24$,
   4. ${}n_{1}=2$, ${}n_{2}=3$, ${}n_{3}=5$, and ${}n=60$.

Proof, Write another proof