Correct Answer: 4 and 64

Solution : Let the two numbers as $a$ and $b$. The mean proportion of two numbers is the square root of their product.
$\sqrt{ab} = 16$
Squaring both sides,
$ab = 256$.....................................(1)
The third proportional of two numbers $a$ and $b$ is the number which is to $b$ as $b$ is to $a$.
$\frac{b}{a} = \frac{1024}{b}$
⇒ $b^2 = 1024a$...............................(2)
We can substitute $a = \frac{256}{b}$ from equation (1) into equation (2).
⇒ $b^2 = 1024×\frac{256}{b}$
⇒ $b^3 = 1024×256$
⇒ $b$ = 64
Substituting $b = 64$ in equation (1)
⇒ $64a = 256$
⇒ $a$ = 4
So, the two numbers are 4 and 64.
Hence, the correct answer is '4 and 64'.