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 amguessingthat 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!