Suppose that Alice$$s RSA public key is $($N$,$ e$)=(33,7)$

a$.$ Find her private key, d$.$

b$.$ If Bob encrypts the message M $=17$ using Alice$$s public key, what is the ciphertext C$?$ Show that Alice can decrypt C to obtain M$.$

c$.$ Let S be the result when Alice digitally signs the message M $=19.$ What is S$?$ If Bob receives M and S$,$ explain the process Bob will use to verify the signature and show that in this particular case, the signature verification succeeds.