Being $2011$ prime, calculate 2009! divided by 2011 I'm working in the following excercise:

Being $2011$ prime, calculate $2009!$ divided by $2011$

By Wilson's theorem I have:
$$2010! \equiv -1 \mod 2011$$
$$2009! * 2010 \equiv -1 * 2010 \mod 2011$$
$$2009! \equiv -2010 \mod 2011 $$
$$2009! \equiv 1 \mod 2011$$
Checking the answer is correct but I'm not sure about step two, is that correct? any help will be really appreciated.
 A: The right side on the second line should still be $-1$ because you just split the product on the left.  Now note that $2010 \equiv -1 \pmod {2011}$ so multiply both sides by $2010$
$$2009! * 2010 \equiv -1 \pmod {2011}\\2009!\cdot 2010^2\equiv -1\cdot 2010 \pmod {2011}\\
2009!\equiv 1 \pmod {2011}$$
You also just removed a factor $2010$ from the left between the second and third lines, which canceled out the error from the first to the second.
A: No.  It's not. $2009*2010 = 2010! \equiv -1 \pmod {2011}$.  Why did you multiply it by $2010$ when you don't know $2009! \not \equiv -1 \pmod {2011}$.
And then when you had $2009!* 2010 \equiv -2010$ you went to $2009! \equiv -2010$ by simply dropping the $2010$ out of nowhere.  You dropped it in for no reason.  And then you dropped it out for no reason.  
That won't work.   Two wrongs frequently make a right and they did in this case.  But they did so for the wrong reasons.
$$\color{blue}{2010!} \equiv \color{blue}{-1}\pmod{2011}$$
$$\color{blue}{2009!*2010}\equiv \color{blue}{-1}*\color{red}{2010} \pmod{2011}$$ (This is wrong!  The blue colors are all equivalent but the red $\color{red}{2010}$ came from absolutely nowhere.)
$$\color{orange}{2009!} \equiv \color{blue}{-}\color{red}{2010}\pmod{2011}$$
(This is wrong!  The $\color{blue}{2009!*2010}$ simply turned into $\color{orange}{2009!}$ for no reason!  What happened to the $\color{blue}{2010}$?  Where did it go?)
....
Instead do:
$2010! \equiv -1 \pmod {2011}$
$2009!*2010 = 2010! \equiv -1 \pmod {2011}$.
Now notice $2010 \equiv -1 \pmod {2011}$ so
$2009!*(-1) \equiv - 1\pmod {2011}$ and 
$2009!*(-1)*(-1) \equiv (-1)(-1) \pmod {2011}$ and so
$2009! \equiv 1 \pmod {2011}$.
