I'm thinking putting it into modulo form: there exists a natural number $n$ for which
$2^{n}\equiv 1 \pmod {11}$
but I don't know what to do next and I'm still confused how to figure out remainders when doing modulos, like $2^n\equiv \;?? \pmod{11}$. Is there some pattern to find $??$ or you would have to use specific numbers for $??$ which is divisible by $11$?