A number theory problem by Anshaj Shukla

$m^2(m+1)^2$ is a multiple of $5$ , and $m^2(m+1)^2\le1000000$ .

So, the only possible values for $m$ is $0,4,5,9,10,14,15,19,20,24,25,29,30$

For every possible $m$ , there is an $n$ fitting the equation. Hence, there are $13$ possible solutions.