Zp question

[yda]

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 ?

[matt grime]

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

[yda]

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

[matt grime]

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.

[yda]

Its clear for me now, thanks