[Number Theory]
Problem: Let $x$ be the smallest positive integer such that $1584 \cdot x$ is a perfect cube, and let $y$ be the smallest positive integer such that $x y$ is a multiple of 1584 . Compute $y$.

Solution: In order for $1584 \cdot x$ to be a perfect cube, all of its prime factors must be raised to powers divisible by 3 . Because $1584=2^{4} \cdot 3^{2} \cdot 11$, $x$ must be of the form $2^{3 k+2} \cdot 3^{3 m+1} \cdot 11^{3 n+2} \cdot r^{3}$, for nonnegative integers $k, m, n, r, r>0$. Thus the least positive value of $x$ is $2^{2} \cdot 3 \cdot 11^{2}=1452$. But in order for $x y$ to be a positive multiple of $1584, x y$ must be of the form $2^{a} \cdot 3^{b} \cdot 11^{c} \cdot d$, where $a \geq 4, b \geq 2, c \geq 1$, and $d \geq 1$. Thus $y$ must equal $2^{2} \cdot 3^{1}=\mathbf{1 2}$.

Final answer: 12