Cubes And Squares

Let $S = 2^{a}\cdot3^{b}\cdot5^{c}$ Then, from the condition that $10S$ is a perfect square, we get $2\mid (a+1), 2\mid b, 2\mid (c+1)$ From the condition that $6S$ is a perfect cube, we get $3\mid (a+1), 3\mid (b+1), 3\mid c$

Combining, we can determine that the lowest possible value of a is 5, b is 2 and c is 3. Plugging in, we get the answer, 36000.