(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that if [itex]p[/itex] is a prime and [itex]a, b \in \mathbf{Z}[/itex] with [itex]a \not \cong 0 \mod p[/itex], then [itex]ax \cong b \mod p[/itex] has a unique solution modulo [itex]p[/itex].

I'm having a hard time proving there exists only one solution by using a contradiction.

But my biggest problem is that I don't understand why this statement should have a unique solution modulo p. For example, let a = 3, b = 2, and p = 2. Then [itex]3x \cong 2 \mod 2[/itex] has multiple solutions. (x = 2,4,6,...) And that's just one example. Or does the "unique modulo p, mean that you take all of the solutions, and apply mod p to them and that is unique? For example (x = 2,4,6,...) mod p = 0.

2. Relevant equations

N/A

3. The attempt at a solution

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# I'm having a hard time showing this congruence has a unique solution modulo p.

**Physics Forums | Science Articles, Homework Help, Discussion**