I need to prove that if

a,b, a is not 0, in Zp than there is n that b = na

 

how should I start proving that ?

Firstly, this isn't true unless p is a prime, if it is a prime then say so.

 

The answer as always follows from the hint: all you know (if p is a prime), is that a and p are coprime. Hence all you can do is use the single result at your disposal, ie Euclid's Algorithm.

 

Let us know where that gets you

well, the prime is not mention in my question but after what u said I agree that its must be.

still I dont know where to start proving that and how can I use Euclid's Algorithm for the solution

What does Euclid tell us? That given a and p coprime, there are integers x and y with

 

ax+py=1

 

 

can you now see how to express b as "a" times something plus something congruent to zero mod p?

 

Note that if all we require is that a is not zero then p must be a prime, but it suffices just for a and p to be coprime.

Its clear for me now, thanks