3 Problems: Modulus, Module/Vector space, and Subgroup Order

With the exception of the last one, I've completed most of the problems myself before coming here; so this shouldn't be too much...

Modulus: 2^155 mod 53 =?
I was able to reduce it to 2^51 mod 53 using Fermat's Theorem. But I'm not sure how to get it down the rest of the way without using a calculator (which says the answer is 27). The Chinese *something* Theorem (sorry; brain has been warped too much tonight) doesn't work since 53 is already a prime number... unless I'm missing something on how to use it. So what Theorem, formula, rain-dance do I use to get the number down to something more manageable?

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Module/ Vector Space

Consider the set A=(a+b*Sqrt(3) | a,b are both real numbers (R))
(Basically; both A and R define the set of all Real Numbers) - we're also given three operators that act like multiplication (for all intents and purposes) and two operators that act identical to addition (for all intents and purposes).

After a series of questions asking about Abelian Groups and Mathematical Rings we come to...

Is {A,R,*,+,*,+,*} a module? Explain.

I'm not as sure of the definition of a module as I am of the abelian group or ring; all the definitions I come across (including my own notes) tend to only confuse me further.

Note: The first multiplication and addition only work with A's setup; the middle multiplication works only when combining both sets; and the last two only work with set R.

I am guessing that it is based on the next question (Is it an unitary module? - Which I can only assume is a yes as well since the basic idea is 1x=x... please correct me if thats wrong.)

Second part;
Is it a vector space? Explain.

I might assume I could get this part if I understood the first; but I might not; since I'm not sure how to relate abelian groups or Rings with vector spaces (if that's possible? -I might assume that only modules could be related).

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Subgroup Order:
I'm outright lost on this one.

I understand everything but what is meant by 'Order 4'
"Find a subgroup H1 of order 4."

I'm given a set consisting of 16 numbers that 'rotate' - a modulus- If I add one to 15, I go not to 16, but 0. This is part of an Abelian group that includes an equivilent form of addition.

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All of the problems are much bigger than this; I've narrowed them down to these parts and simplified them further in order to explain it.
Although I have a few more problems than just these; I have chosen problems I believe will allow me to answer everything else once I understand them.

I apologize if my thought process simplified them too much (or even too little! >.>), but thank you in advance for any assistance you are able to provide!

First one:
You were on the right track. Fermat's theorem is the way to go. We know that 2^52 = 1 (mod 53). 155 = 51 + 2*52 (as you mentioned) so 2^155 = 2^51 (mod 53), and 2^51 = (2^52)/2... So 2^51 = (2^52)/2 (mod 53) = 1/2 (mod 53) = 54/2 (mod 53) (since 54 = 1 (mod 53)) = 27 (mod 53).

Second one:
Sorry I'm lost.

Third one:
Order is the same as the number of elements in a group. So you should find a subgroup with 4 elements. If you're given a generator, g, of the original group (the one with 16 elements, of order 16), a typical generator for a subset of order 4 is g^4... Since (g^4)^4 = g^16 = 1, the identity so your subgroup would be g^4, (g^4)^2 = g^8, (g^4)^3 = g^12, (g^4)^4 = g^16 = 1.