Show that the number $3^{341}-3$ isn't divisible by $341$.
We've just covered Fermat's little theorem and linear congruences in my Algebra class.
I've realized that $341 = 11*31$ and I've wrote down:
$3^{341}-3 \equiv mod$ $341 $
How can I isolate the two prime numbers from the modulo? I'm also aware that $3^{341}$ could be written as $3^{31^{11}}$ but the $mod$ $341$ is keeping me stuck.
Hint: $341\mid 3^{341}-3$ if and only if $$11\mid3^{341}-3\quad\hbox{and}\quad 31\mid3^{341}-3\ ,$$ so you need (and only need) to show that one of these statements is false. Actually one of them is true, so you need to look at the one which isn't, but this is still not very hard using Fermat.