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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