$3^-1 mod 59$

how do negative exponents and modular arithmetic work?

Question

$3^-1 mod 59$

how do negative exponents and modular arithmetic work?

jim_thompson5910 · Answer

3^-1 is the multiplicative inverse of 3 in mod 59

so you are looking for an x value that makes this equation true

3x = 1 (mod 59)

bibby · Answer

ah, so 3^-5 would be 3^5x?

bibby · Answer

what does it mean when it says $3^{-5}\cong 20$

bibby · Answer

mod 59, but that isn't important (I don't think)

ganeshie8 · Answer

we know that $3^{-1}$  is  $\dfrac{1}{3}$
in modulo $59$ : 

$$\dfrac{1}{3} = \dfrac{20}{60}\equiv  \dfrac{20}{1} = 20$$

bibby · Answer

oh yeah, you showed me this trick a week ago

bibby · Answer

and so $3^{-5}=(3^{-1})^5$

bibby · Answer

which is congruent to 17, which is what my professor got.

awesome. thank you so much ganesh. sorry about being a jerk jim

bibby · Answer

and the other 2