The correct answer is (c).
The geometric mean of a set of numbers is the product of the numbers, taken to the power of $1/n$, where $n$ is the number of numbers in the set. In other words, if we have a set of numbers $x_1, x_2, \dots, x_n$, then the geometric mean is:
$$G = \sqrt[n]{x_1 x_2 \dots x_n}$$
The geometric mean of a set of numbers is always greater than or equal to the arithmetic mean of the numbers. This is because the geometric mean takes into account the product of the numbers, while the arithmetic mean only takes into account the sum of the numbers.
In the question, we are given that $X_1, X_2, \dots, X_n$ have a geometric mean of $G_1$ and $Y_1, Y_2, \dots, Y_n$ have a geometric mean of $G_2$. We are then asked to find the geometric mean of $Z_1, Z_2, \dots, Z_n$, where $Z_i = X_i / Y_i$.
We can write the geometric mean of $Z_1, Z_2, \dots, Z_n$ as:
$$G_Z = \sqrt[n]{Z_1 Z_2 \dots Z_n} = \sqrt[n]{X_1 X_2 \dots X_n / Y_1 Y_2 \dots Y_n} = \sqrt[n]{G_1 / G_2} = \sqrt{G_1 / G_2}$$
Therefore, the geometric mean of $Z_1, Z_2, \dots, Z_n$ is $\sqrt{G_1 / G_2}$.
Option (a) is incorrect because $G_1$ and $G_2$ are the geometric means of $X_1, X_2, \dots, X_n$ and $Y_1, Y_2, \dots, Y_n$, respectively, and the geometric mean of a set of numbers is always greater than or equal to the arithmetic mean of the numbers. Therefore, $G_1$ and $G_2$ cannot be the geometric mean of $Z_1, Z_2, \dots, Z_n$.
Option (b) is incorrect because $G_1 / G_2$ is the ratio of the geometric means of $X_1, X_2, \dots, X_n$ and $Y_1, Y_2, \dots, Y_n$, and the geometric mean of a set of numbers is always greater than or equal to the arithmetic mean of the numbers. Therefore, $G_1 / G_2$ cannot be the geometric mean of $Z_1, Z_2, \dots, Z_n$.
Option (c) is correct because the geometric mean of $Z_1, Z_2, \dots, Z_n$ is $\sqrt{G_1 / G_2}$.
Option (d) is incorrect because the geometric mean of $Z_1, Z_2, \dots, Z_n$ is $\sqrt{G_1 / G_2}$.