Alice and Bob wish to use the Diffie$-$Hellman key exchange. They agree to use the prime $37$ and base $g=5.$ Bob's private keys is $k_B=5.$ Alice sends her public key $A=16$ to Bob.

$($a$)$ Is $g$ primitive root modulo $37?$

$($b$)$ Is it necessary for $g$ to be a primitive root to make the Diffie$-$Hellman key exchange protocol work?

$($c$)$ What is Bob's public key $B?$

$($d$)$ Compute the shared secret key.