Least Positive Residue How would I find the least positive residue of
say
$6! \bmod 7$
or
$12! \bmod 13$
I just learned modular arithmetic and my book doesn't explain what least positive residues are so I'm a bit lost.

Do you already know this?:  $a\equiv b \bmod m \implies ac\equiv bc \bmod m$.
You can use this to keep the numbers small as you investigate these results:
$4! = 1\cdot 2\cdot 3\cdot 4 \equiv 24\equiv 3 \bmod 7$
$5! = 4!\cdot 5 \equiv 3\cdot 5 \equiv 15 \equiv 1 \bmod 7$
$6! = 5!\cdot 6 \equiv 1\cdot 6 \equiv 6 \bmod 7$
(and to reassure yourself that this is working, $6!=720 \equiv 20\equiv 6 \bmod 7\ \  \checkmark$)
We can shortcut a little with some precalculation for the $\bmod 13$ question:
$2\cdot 7=14\equiv 1\bmod 13$,
$3\cdot9=27\equiv 1\bmod 13$,
$4\cdot 10 = 5\cdot 8=40\equiv 1 \bmod 13$,
giving us:
$10! \equiv 1^5\cdot 6\equiv 6 \bmod 13$  
and so on.
