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

If $p$ is prime, then you have $p$ divides $2^{p-1} - 1$ by Fermat's little theorem, which would show that your result holds if $121$ is prime. However you have that $121 = 11^2$ is not prime, so it's more complicated. A simple calculation shows that the congruence indeed does not hold.
