a. p=5, q=13
n=p*q=65
$\Phi (n)$=(p-1)*(q-1)=48
d(private key) should be chosen such that d and $\Phi (n)$ have no common factors(they are relatively prime)
Possible choices for d=5,7,11,13,17
b. p=5, q=31 and d=37
n=p*q=155
$\Phi (n)$=(p-1)*(q-1)=120
the following relation must hold
e*d=1 MOD $\Phi (n)$
e*37=1 MOD 120
e*37 could be 121, 241, 361, 481 and so on
for here we obtain e=13
c. p=3, q=11, d=9
n=p*q=33
$\Phi (n)$=(p-1)*(q-1)=20
e*d=1 MOD $\Phi (n)$
e*9=1 MOD 20
e=9
for encryption $C=P^e MOD$ n
hello=8, 5, 12, 12, 15
h => $8^9 MOD$ 33=29
e => $5^9 MOD$ 33=20
l => $12^9 MOD$ 33=12
o => $15^9 MOD$ 33=3