Do identical, equally spaced lenses possess the same magnification, or does it depend on the sample's distance? Do identical, equally spaced lenses always have the same magnification, or does it depend on the sample's distance to the first lens?
This question comes from trying to solve:
"There are 4 identical and equally spaced lenses between the sample and the camera. The distance from the first to the last lens is 70cm. Find their individual magnifications to achieve a magnification of $10^6$."
I know that when there's more than a lense, the total magnification is the product of the magnification values for each lense, meaning $M_{total}= M_1 M_2 ...M_n$ .
This makes me think that, because they are identical, the magnification of each of these lenses must be $(10^6)^{1/4} =10 \sqrt{10} \approx 31.62$
This does not use the distance between each lens which makes me think it is not correct. But if I think of using this distance then I must require the initial distance from the sample to the first lens, is this not so? Is there a lack of information in this exercise?
Does the magnification solely depend on its shape/material (in which case they would all be the same)?
 A: "Does the magnification solely depend on its shape/material (in which case they would all be the same)?"
In the realm of simple geometrical optics, the magnification of a lens is given by q/p, where q is the image distance and p is the object distance. So it's not really a property of the lens but rather of the image conjugates in any particular case. (A magnifier might be characterized by a magnification but that type of lens is always used in a particular way).
The only way the magnification of each identical lens can be the same is if q/p is the same for each one. Let's say the image and object distances for the first lens are q$_1$ and p$_1$. One way for the second lens to have the same magnification is if the object distance to the second lens, p$_2$, equals p$_1$. Etc. for the remaining two lenses. Under this special case, in which each lens has the same object and image distances, I think your result would be correct.
But I'm not totally clear on what the assumptions of the problem are.
A: I suspect the purpose of this problem is to see if you know how to use the lensmaker's formula, rather than to design a practical lens system.  So IMHO the right approach would be to construct a formula for total magnification, with the given parameters as constraints, set magnification at $10^6$, then solve for the lens spacing as a function of lens focal length.  Given the focal length of the lens, you can go back to the lensmaker's formula to determine lens curvature as a function of refractive index.  IF there is a value of refractive index between 1 and, say, 2 that works for a curvature radius less than half the lens spacing, you've solved the problem.
