How can we find Public key have only 8 or 16bits? How many messages does Eve need to know the Public key in RSA? Suppose Alice sends messages to Bob by encrypting the messages with Bob's public key.

Eve knows that the data is encrypted using RSA, but does not know the public key. Can Eve figure out the public key just by observing the encrypted messages?

And if so, approximately how much data would it take for Eve to discover the public key?

If we know that the public key has only 16bits.

It's have over 6500 primes smaller than 2^16. How long we can find the public key ??
speed : 1.000.000 cal/s.
If the plaintext is easily recognizable, one message is sufficient. Simply brute force all 16 bit RSA keys, decrypt the ciphertext. If the result "looks" like plaintext, you have found the key. A 16 bit RSA keyspace should be easy to brute force.