Prove that there no positive integral solution to this equation. Prove that there doesn't exist positive integers $a,b,c,n$ such that this equality holds:
$6(6a^2+3b^2+c^2)=5n^2$
I found reduced the equation as follows:
$2a^2+b^2+3m^2=10r^2$
But any mod upto $7$ is not useful here. 

It is clear $b+m$ is even,
case 1: if $b,m$ are odd numbers,then 
$$b^2\equiv 1\pmod 8,3m^2\equiv 3\pmod 8,2a^2\equiv 0\textbf{or}2\pmod 8$$
so
$$LHS=2a^2+b^2+3m^2\equiv 4\textbf{or} 6\pmod 8$$
but $$RHS=10r^2=0\textbf{or} 2\pmod 8$$
a contradiction.
case 2: if $b,m$ are even number,then let
$b=2b',m=2m'$,so
$$a^2+2b'^2+6m'^2=5r^2$$
and same as case 1 methos,we have $a,r$must even numbers,
so let
$a=2a',r=2r'$,then we have
$$2a'^2+b'^2+3m'^2=10w'^2$$
but $w'<w$,then use Proof by infinite descent a contradiction.
