Find the smallest number $y$ such that $y \times 162$ is a perfect cube.

Find the prime factorization of $162$:

For the product $y \times 162$ to be a perfect cube, the exponent of every prime factor must be a multiple of $3$.

• For the prime factor $2$: current power is $1$. The smallest multiple of $3$ greater than or equal to $1$ is $3$, so we need $2^{3 - 1} = 2^2 = 4$.

• For the prime factor $3$: current power is $4$. The smallest multiple of $3$ greater than or equal to $4$ is $6$, so we need $3^{6 - 4} = 3^2 = 9$.

Thus, the smallest positive integer $y$ is:

Verification: $36 \times 162 = 5832 = 18^3$, which is a perfect cube.

Hence, the correct option is (D).