Division algorithm and unique Gaussian integers

[Proggy99]

Homework Statement

Theorem
Let \alpha\neq0 and \beta be Gaussian integers. Then there are Gaussian integers \tau and \rho such that \beta=\tau\alpha+\rho and N\left(\rho\right)<N\left(\alpha\right)

Problem
Show that the Guassian integers \tau and \rho in the Theorem are unique if and only if \beta is a multiple of \alpha

Homework Equations

The Attempt at a Solution

Can someone please give me a jumping off point for this question because I am not sure how to proceed? I used the theorem in previous problems to actually solve for \tau and \rho given \beta and \alpha, but I am not sure how to go about this problem. Thanks

 

[Proggy99]

Homework Statement

Theorem
Let \alpha\neq0 and \beta be Gaussian integers. Then there are Gaussian integers \tau and \rho such that \beta=\tau\alpha+\rho and N\left(\rho\right)<N\left(\alpha\right)

Problem
Show that the Guassian integers \tau and \rho in the Theorem are unique if and only if \beta is a multiple of \alpha

Homework Equations

The Attempt at a Solution

Can someone please give me a jumping off point for this question because I am not sure how to proceed? I used the theorem in previous problems to actually solve for \tau and \rho given \beta and \alpha, but I am not sure how to go about this problem. Thanks

I did some poking around trying to find hints but can not seem to find any beyond proving uniqueness for the division algorith for real numbers. I can not seem to find a way to apply a similar method here. Help please.