Abstract Algebra modular Arithmatic Proof

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Homework Statement



Prove 10n ≡18 10 for all n ϵ N

Homework Equations




I have no idea where to even begin this proof.

The Attempt at a Solution


 
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What is the definition of a\equiv_n b ?
 
it is said to be congruent integer modulo if n|a-b
 
So apply this definition to your problem. What are you trying to show? I.e., you are trying to show ____ is divisible by ____?

What would have to be true for a number to be divisible by ____?
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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