is 121 divides the my pattern for base 2? Is $121|2^{120}-1$? If yes, is there any online free calculation to check these type of values?
Advanced thanks to one and all!
-Richard Sieman

Since $ \phi(121) = 110$, we know that $ 2^{110} \equiv 1 \pmod{121}$ by Euler's Theorem. 
Hence, $2^{120} \equiv 2^{10} \equiv 1024 \equiv 56 \pmod{121}$, and thus $ 2^{120} -1\equiv 55 \pmod{121}$.
