GATE CS 2017 Set 1 QGA-4 — General Aptitude

To find the smallest positive integer $y$ such that $y \times 162$ is a perfect cube, we first find the prime factorization of $162$:

For a number to be a perfect cube, the exponent of each prime factor in its prime factorization must be a multiple of $3$.

Let $y = 2^a \times 3^b$. Then:

To make the exponents multiples of $3$ with the smallest non-negative integers $a$ and $b$:

• For the base $2$: The exponent is $a + 1$. The smallest multiple of $3$ greater than or equal to $1$ is $3$.
  So, $a + 1 = 3 \implies a = 2$.
• For the base $3$: The exponent is $b + 4$. The smallest multiple of $3$ greater than or equal to $4$ is $6$.
  So, $b + 4 = 6 \implies b = 2$.

Thus, the smallest value of $y$ is:

Therefore, the correct option is D.