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.