278. Find two numbers whose product is a three-digit number that is the cube of some number, and the quotient is the square of this number.
278. From the system $\left\{\begin{array}{l}x y=t^{3} \\ \frac{x}{y}=t^{2}\end{array}\right.$ we find $x^{2}=t^{5}, y^{2}=t$;

thus, $t$ must be a perfect square; but since by the condition $x y$ is a three-digit number, then $100<t^{3}<1000, 5 \leqslant t \leqslant 9$. Therefore, $t=9, x=243, y=3$.

Answer. 243 and 3.