Respuesta :
Answer: 2
==========================================================
Explanation:
Note that 2^4 = 16 is one less than 17.
Because of this nature of being 1 less, we can say the remainder of 16/17 is -1 because it comes up one short.
In terms of modular arithmetic, we can say,
[tex]2^4 \equiv 16 \ (\text{mod } 17)\\\\2^4 \ \equiv -1 (\text{mod } 17)\\\\[/tex]
Then let's raise both sides to the 20th power. I'm picking 20 because 20*4 = 80 which will help build toward 81
[tex]2^4 \equiv -1 \ (\text{mod } 17)\\\\\left(2^4\right)^{20} \equiv (-1)^{20} \ (\text{mod } 17)\\\\2^{20*4} \equiv 1 \ (\text{mod } 17)\\\\2^{80} \equiv 1 \ (\text{mod } 17)\\\\[/tex]
Lastly, we'll multiply both sides by 2 to get to the final answer
[tex]2^{80} \equiv 1 \ (\text{mod } 17)\\\\2*2^{80} \equiv 2*1 \ (\text{mod } 17)\\\\2^{1}*2^{80} \equiv 2 \ (\text{mod } 17)\\\\2^{1+80} \equiv 2 \ (\text{mod } 17)\\\\2^{81} \equiv 2 \ (\text{mod } 17)\\\\[/tex]
So if we were to divide 2^81 over 17, then we get some quotient and a remainder of 2
