Question 59

Find the prime factorization of 162:

For the product $y \times 162$ to be a perfect cube, the exponent of every prime in its prime factorization must be a non-negative multiple of 3 (that is, in $\{0, 3, 6, 9, \dots\}$).

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

• For the prime 2: the current exponent is 1. The smallest exponent $\ge 1$ that is a multiple of 3 is 3. Thus, $1 + a = 3 \implies a = 2$.
• For the prime 3: the current exponent is 4. The smallest exponent $\ge 4$ that is a multiple of 3 is 6. Thus, $4 + b = 6 \implies b = 2$.

To minimize $y$, set any other prime factors $k = 1$:

Verification:

Thus, the smallest positive integer $y$ is 36.