$(20$pts$)$ Let RSA modulus $n=35$ and public key $e=7.$

$($a$)$ Can you figure out the private key $d?$

$n=5**7.$

$(n)=(5-1)*(7-1)=24$

$7**dmod24=1.$ Then $d=7.$

$($b$)$ Let message $M=2,$ encrypt $M$ using the RSA parameters above.

$C=2^7$mod$35=23$