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

You can rewrite your expression as$$2^{120}-1\\=(2^{60}+1)(2^{60}-1)\\=(2^{60}+1)(2^{30}+1)(2^{30}-1)\\=(2^{60}+1)(2^{30}+1)(2^{15}+1)(2^{15}-1)$$
so $$2^{120}-1\pmod{ 121}\\\equiv(2^{60}+1)(2^{30}+1)(2^{15}+1)(2^{15}-1)\pmod{ 121}$$
but since $2^{15}-1=32767 \equiv 97 \pmod{ 121}$
$$2^{120}-1\pmod{ 121}\\\equiv(2^{60}+1)(2^{30}+1)(2^{15}+1)(2^{15}-1)\pmod{ 121}\\\equiv(2^{60}+1)(2^{30}+1)\cdot97\cdot99\pmod{ 121}\\\equiv(2^{60}+1)\cdot44\cdot46\pmod{ 121}\\\equiv90\cdot88\pmod{ 121}\\\equiv55\pmod{ 121}$$
