$(1)$ When using RSA encryption $($with parameters $n=p*q,e,d)$ sometimes

you will meet a scenario when the ciphertext $c$ and the plaintext $m$ coincide.

What is the possible number of such coincidences?

$(2)$ Let $n=p*q$ be modulus and $e$ be public exponent. Suppose there exists a

polynomial algorithm that given a ciphertext $c$ it computes the second bit of

the binary expansion of the plaintext $m,$ i$.$e$.f(c)={}_1,$ where

$m={}_0+{}_1*2+{}_2*2^2+$dots$+{}_k*2^k.$

Is it possible to restore $m$ in polynomial time?

$(3)$ Professor X$.$ publishes his RSA public key: $n=176493928789$ and exponent

$e=3.$ Then, he sends a series of encrypted messages:

$c_1=151478500367,c_2=83888564074,c_3=56766452018,$

$c_4=101396164208,c_5=72405022489.$

If you know that messages are encrypted over the English alphabet, can you

restore original messages?