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 3^{16}\equiv 1 \pmod {17}



Also: 3^{2012} = (3^{16})^{125}*3^{12} \pmod{17}



So I am left with 3^{12}\pmod{17}.




Again I'm going to use fermat's theorem so: 3^{12} = \frac{3^{16}}{3^{4}} \pmod{17}



Here I am stuck because I get 3^{-4} \pmod{17} and I don't know how to calculate this because I don't know what \frac{1}{81} \pmod{17} is.



I know 81 = 13 \pmod{17}



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


Answer



3^{12}=(3^3)^4=10^4 (mod 17), so we have to find 10000 (mod 17), which is evidently 4 (mod 17).



No comments:

Post a Comment

real analysis - How to find lim_{hrightarrow 0}frac{sin(ha)}{h}

How to find \lim_{h\rightarrow 0}\frac{\sin(ha)}{h} without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...