# 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.

• It's perfectly correct. Jan 30, 2019 at 22:21
• No. It's not $2009*2010 = 2010! \equiv -1 \pmod {2011}$. Why did you multiply it by $2010$ when you don't know $2009! \not \equiv -1 \pmod {2011}$. Jan 30, 2019 at 22:23
• It should be noted that 2011 is prime Jan 30, 2019 at 22:41
• Perhaps a lemma to wilsons thereom should be: For $p > 2$ a prime. $(p-1)! \equiv -1 \pmod p$ so $(p-2)! = -(p-2)!(-1) \equiv -(p-2)!(p-1)=-(p-1)! \equiv -(-1) \equiv 1 \pmod p$. Jan 30, 2019 at 22:49

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.

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?)

....

$$2010! \equiv -1 \pmod {2011}$$
$$2009!*2010 = 2010! \equiv -1 \pmod {2011}$$.
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
$$2009! \equiv 1 \pmod {2011}$$.