A number with digits $2$ and $5$ is divisible by $2^{2005}$

Question

I found an old problem on a website but no answer is given. Here is the problem :

Prove that there is a unique positive integer consisting entirely of digits $2$ and $5$, having exactly $2005$ digits and divisible by $2^{2005}$.

Here is where i am : it must finish with a $2$, but i have absolutely no other clue.