polynomial division problem ( find the remainder) How to find out what remainder will $\,(x-1)^{2013}\,$ have upon division by $\,x^2-2x+2\,?\;$ 
I've never solved anything like this before, so I have no ideas at all.. 
Thanks in advance! 

Hint: Upon division by $x^2 - 2x + \color{blue}{\bf{1}}$, there is no remainder: that is $$\frac{(x-1)^{2013}}{x^2 - 2x + 1} = \frac{(x - 1)^{2013}}{(x-1)^2} = (x - 1)^{2011}$$
Note: $\quad(x-1)^2 + 1 \equiv 0\pmod{x^2-2x+2}\;\implies\; (x-1)^2 \equiv -1 \pmod {x^2-2x+2}$ 
So what does this mean for 
$
\begin{align}
(x - 1)^{2013}\pmod{x^2 - 2x +2} &= (x-1)(x-1)^{2012}\pmod{x^2 - 2x +2} \\ &= (x-1)[(x-1)^2]^{2012/2}\pmod{x^2 - 2x +2}\\ &= (x-1)[(x-1)^2]^{1006}\pmod{x^2 - 2x + 1}\quad?\end{align}$
