AMC 12 Mock Exam 3 Problem 2

The total number of unit cubes is $2025n$, so we need the smallest $n$ for which $2025n$ is a perfect cube. Since $2025=3^4\cdot5^2$, a multiple $2025n$ is a perfect cube exactly when every prime exponent in it is a multiple of $3$. The exponent of $3$ must rise from $4$ to at least $6$ and the exponent of $5$ from $2$ to at least $3$, so $n$ must be a multiple of $3^2\cdot5=45$; and $2025\cdot45=3^6\cdot5^3=(9\cdot5)^3$ is a cube. The answer is $\boxed{45}$.