Thursday, January 31, 2019

discrete mathematics - How to compute 32003pmod99 by hand?





Compute 3^{2003}\pmod {99} by hand?





It can be computed easily by evaluating 3^{2003}, but it sounds stupid. Is there a way to compute it by hand?


Answer



I would calculate separately modulo 9 and 11 and put the pieces together at the end.



Modulo 9 is trivial, we get 0.



Note that 3^5\equiv 1\pmod{11}, so 3^{2000}\equiv 1\pmod{11}, and therefore 3^{2003}\equiv 3^3\equiv 27\pmod{11}. This is already congruent to 0 modulo 9, so we are finished.


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