Q$3.$ Alice and Bob use the Diffie$-$Hellman key exchange technique with a common prime $q=23$ and a primitive root $=5.$

a$)$ If Bob has a public key $Y_B=10,$ what is Bob's private key $x_B?$

b$)$ If Alice has a public key $Y_A=8,$ what is the shared key $K$ with Bob?

c$)$ Show that $5$ is a primitive root of $23.$