Wednesday, 9 August 2017

number theory - How can I calculate the remainder of 32012 modulo 17?



So far this is what I can do:



Using Fermat's Little Theorem I know that 3161(mod17)



Also: 32012=(316)125312(mod17)



So I am left with 312(mod17).




Again I'm going to use fermat's theorem so: 312=31634(mod17)



Here I am stuck because I get 34(mod17) and I don't know how to calculate this because I don't know what 181(mod17) is.



I know 81=13(mod17)



But I know the answer is 4. What did I do wrong?


Answer



312=(33)4=104 (mod 17), so we have to find 10000 (mod 17), which is evidently 4 (mod 17).



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