$($Modification of exercise $36$ in section $2\mathrm{.}5$ of Rosen.$)$

The goal of this exercise is to work thru the RSA system in a simple case:

We will use primes $p=41,q=53$ and form $n=41*59=2178.$

$[$This is typical of the RSA system which chooses two large primes at random generally, and multiplies them to find $n.$ The public will know $n$ but $p$ and $q$ will be kept private.$]$ Next we encode letters of the alphabet numerically say, via the usual:

$)=0,B=1,C=2,D=3,E=4,F=5,G=6,H=7,1=8,$

$J=9,K=10,L=11,M=12,N=13,O=14,P=15,Q=16,R=17.$

$S=18,T=19,U=20,V=21,W=22,x=23,Y=24,Z=(25$

We will practice the RSA encryption on the single integer $15.($which is the numerical representation for the letter $"$P$").$ In the language of the book, $M=15$ is our original message.

The coded integer is formed via $c=M^c$modn.

Thus we need to calculate ${15}^{17}$mod$2173.$

This is not as hard as it seems and you might consider using fast modular multiplication.

The canonical representative of ${15}^{17}$mod$2173$ is