Problem 2

The left side is divisible by $6$, so $6\mid5n^2$. Since $5$ is coprime to $6$, both $2$ and $3$ divide $n^2$, hence $6\mid n$. Put $n=6n_0$; after cancellation the equation becomes $6a^2+3b^2+c^2=30n_0^2$.