Question

x is the smallest number such that $\frac{x}{2}$ is a perfect square and $\frac{x}{3}$ is a perfect cube. Then, the number of divisors of x is -

• A. 12
• B. 16
• C. 20
• D. 24

Answer 1

The correct option is C 20

$\frac{x}{2}$ is a perfect square $\Rightarrow \frac{x}{2}=a^2.\frac{x}{3}$ is a perfect cube $\Rightarrow \frac{x}{3}=b^3.\frac{x}{2}\times \frac{x}{3}=a^2\times b^3\Rightarrow x=\sqrt{6a^2{b}^3}=ab\sqrt{6b}.$
For x to be a natural number; 6b has to be a perfect square $\Rightarrow b=6\Rightarrow x=2^3\times 3^4\Rightarrow$ Number of divisors $=20$