Remainder question with $6!$ and 7 Find the remainder when $6!$ is divided by 7.
I know that you can answer this question by computing $6! = 720$ and then using short division, but is there a way to find the remainder without using short division?
 A: As $7$ is prime, use Wilson's Theorem  $$(p-1)!\equiv-1\pmod p$$ for prime  $p$
Now, $\displaystyle -1\equiv p-1\pmod p$
A: Hint $\ $ In analogy with Gauss's trick (see below), to simplify the product we pair up each number with its (multiplicative) inverse mod $7.\,$ Thus $$ 6! = 1\cdot (\overbrace{2\cdot 4}^{\equiv \,1})(\overbrace{3\cdot5}^{\equiv\, 1})\cdot 6 \equiv 1\cdot 1\cdot 1\cdot 6\equiv6\pmod{7}$$
Remark $\ $ This method of pairing up inverses works for any prime - see Wilson's Theorem.
Below is Gauss's trick, imported from a deleted question
$\qquad\qquad \begin{array}{rcl}\rm{\bf Hint}\quad\quad\ \ S &=&\rm 1 \ \ \ +\ \ \: 2\ \ \ \ +\ \:\cdots\  +\ n\!-\!1\ +\ n \\  
\rm S &=&\rm n \ \ +\   n\!-\!1\ +\,\ \cdots\ +\,\quad 2\ \ \ +\ \ 1\\
\hline \\
\rm Adding\ \ \ \ 2\: S &=&\rm n\ (n\!+\!1)\end{array}$  
A famous legend says Gauss used this trick to quickly compute $ 1+2+\:\cdots\:+100\ $ in grade school.
This trick of pairing up reflections around the average value is a special case of exploiting innate symmetry - here a reflection or involution. It's a ubiquitous powerful technique, e.g. see my post on Wilson's Theorem and it's group theoretic generalization.
