What Connections Are There Between Number Theory And Partial Differential Equations?

Number Theory: What is the least residue [math]\mod N = 95[/math] of [math]3^{1.1 \cdot 10^{43}}[/math]?

• This is a practice problem. Since [math]5 \cdot 19[/math] are prime factors of [math]95[/math] I tried to break it into two congruence equations and use CRT, but I can't seem to work this out. By fermat's little theorem we have [math]3^5[/math] congruent to [math]3 \mod (5)[/math], but squaring that multiple times cannot be the solution because then we have to square the RHS of the equation, which is some ridiculous thing. Also [math]3^{19}[/math] congruent to [math]3 \mod 19[/math], which doesn't help much (I think?) and [math]3^{18} [/math] congruent to [math]1 \mod 19[/math] which also doesn't help much (I think...). Some help would be appreciated.

• Answer:

  You want to use the generalization of Fermat's Little Theorem sometimes known as Euler's Theorem.  3 and 95 are relatively prime, so 3 generates the entire multiplicative group (mod 95).  Calculate the order of this group to get the multiplicative order of 3 (mod 95).