jmc

We begin by breaking down $6^{2}10^2$ and $15^4$ into prime factors. Thus we are looking for $$\begin{array}{rl}\gcd (6^{2}10^2,15^4)& =\gcd (2^2{3}^2\cdot 2^2{5}^2,3^4{5}^4)\\ & =\gcd (2^4{3}^2{5}^2,3^4{5}^4).\end{array}$$To build the greatest common divisor of two integers with known prime factors, we take the smaller power of each prime: $$\gcd (2^4{3}^2{5}^2,3^4{5}^4)=3^2{5}^2=15^2=\boxed{225}.$$