- #1

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## Homework Statement

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.

## Homework Equations

N/A