Problem:
Two solid spheres of the same metal but of mass M and 8M fall simultaneously on a viscous liquid, and their terminal velocities are v and nv, respectively. Find the value of n.

Solution:
To solve this problem, we need to understand the concept of terminal velocity and the factors affecting it.

Terminal Velocity:
Terminal velocity is the maximum velocity reached by an object when the drag force due to the surrounding medium is equal to the gravitational force acting on the object. At terminal velocity, the net force on the object is zero, resulting in a constant velocity.

Factors Affecting Terminal Velocity:
There are several factors that affect the terminal velocity of an object falling through a fluid:
1. Mass of the object: Heavier objects experience a greater gravitational force and thus reach a higher terminal velocity.
2. Shape and size of the object: Objects with a larger surface area experience more drag and have a lower terminal velocity.
3. Viscosity of the fluid: Higher viscosity fluids offer more resistance to the motion of objects, resulting in a lower terminal velocity.
4. Density of the fluid: Objects falling through denser fluids experience a greater drag force and have a lower terminal velocity.

Analysis:
In this problem, we are given two solid spheres of the same metal but with different masses (M and 8M) falling through a viscous liquid. Let's assume the terminal velocity of the sphere with mass M is v and the terminal velocity of the sphere with mass 8M is nv.

We can apply the concept of terminal velocity to solve this problem. Since both spheres are made of the same metal, they have the same shape and size. Therefore, the only factor affecting their terminal velocities is their mass.

Solution Steps:
1. The terminal velocity of an object is directly proportional to its mass. Since the terminal velocity of the sphere with mass M is v and the terminal velocity of the sphere with mass 8M is nv, we can write the equation:
v ∝ M (Equation 1)
nv ∝ 8M (Equation 2)

2. Dividing Equation 2 by Equation 1, we get:
nv/v ∝ 8M/M
n ∝ 8
n = 8

Conclusion:
The value of n is 8. Therefore, the correct answer is option (C) 4.