Square numbers

$20n$ can be rewritten as $5*2^2*n$ . For this to be a square number n has to be on the form $5k^2$ since $20n$ is now $5^2*2^2*k^2$ which is of course a square number. Substituting $n=5k^2$ into the other expression: $5*5k^2+275=5^2*(k^2+11)$ . For the whole expreesion to be a square number $k^2+11$ has to be a square number since $5^2$ is already a square number: $k^2+11=j^2 ->11=j^2-k^2 -> 11=(j+k)(j-k)$ . The only whole number solutions are k=5 and j=6. If k =5 then $n=5*5^2=125$ This is the only solution so the sum of all solutions is 125

If $20n$ is a square, then $n$ has to be of the form $5a^2$ . Then

$5n+275 = 25(a^2+11)$

which means $a^2 + 11$ has to be a square. The only two squares that can have a difference of $11$ is $25, 36$ , so $a=5$ , and $n=125$ as the only solution.