As $\frac {11}{6} $ gives remainder of $5$ or $-1$ . we take $-1$ as it eases our calculation .
1) $ \frac{11^{13}}6 $ gives remainder $-1$ .
2) $ \frac{1}6 $ gives remainder 1
FIND : RESULT $1$ - RESULT $2$
$-1 -1 = -2$ which is equal to $4 $ as $6-2 =4$
So remainder is $4$ . It means it is definitely NOT DIVISIBLE by 6 .
.You can check your answers in one of the most trusted sites : wolframalpha , where you can do such calculation involving large numbers . I have done the following for you in the link below . http://www.wolframalpha.com/input/?i=11%5E13%E2%88%921+mod+6
Even scientific calculators in computers perform, mod operation of such large numbers with ease .
These computing sites give confidence on our calculation.