RSA involves two keys: a public key $($e$,$ n$)$ and a private key $($d$,$ n$).$ Suppose p $=5$ and q $=11.$

$($a$)$ Calculate n and $($n$).$ Herein $($n$)$ is Euler$$s totient function which counts the positive integers up to a given integer n that are relatively prime to n$.$

$($b$)$ Choose a public exponent e $=3.$ Verify that e is valid. $($c$)$ Compute the private key d$.$ Show all calculations.