Finding a Complex function

Hello,

I have the following question:

A complex function f is differentiable in a region $\displaystyle \Omega \subset \field{C}.$ and satisfies the equation $\displaystyle |f(z)|=5$ for all $\displaystyle z\in \Omega$. Determine the function f.

I started by saying $\displaystyle f = u + iv$. We know $\displaystyle u^2+v^2 = 25$.

And we know also the Cauchy-Riemann equations since f is differentiable. I tried taking derivatives in $\displaystyle u^2+v^2 = 25$ but I was stuck. An intuitive function to have is $\displaystyle f = 5c$ where c is a complex number with magnitude 1.

Any help is appreciated.

Thanx