Square numbers and multiplication

$1369 \times 1936 = 2650384$

Let $2650384 = m^2 n^2$ , where $m^2$ and $n^2$ are the perfect squares we are looking for. Since $2650384 = 2^4 \times 11^2 \times 37^2$ , $\implies mn = 2^2 \times 11 \times 37 = 1628$ . We need only to consider the divisors of $1628$ . Since $1628$ has $(2+1)(1+1)(1+1) = 12$ divisors, there is only $6$ pairs of $(m,n)$ and $6$ pairs of $(m^2, n^2)$ , which are as follows:

$\begin{array} {|r|r|r|r|} m & n \ \ & m^2 \ & n^2 \quad \\ \hline 1 & 1628 & 1 & 2650384 \\ 2 & 814 & 4 & 662596 \\ 4 & 407 & 16 & 165649 \\ 11 & 148 & 121 & 21904 \\ 22 & 74 & 484 & 5476 \\ 37 & 44 & \blue{1369} & \blue{1936} \end{array}$

Therefore the answer is $\boxed{1369}$ .