Distinguish between cipher-texts of text-book RSA and RSA/OAEP Is there is a possibility to identify from cipher-text that, this is been encrypted using a text-book RSA or RSA with OAEP padding. (By looking at the cipher-text only) ?
We can distinguish between ciphertext of textbook RSA and RSA/OAEP when textbook RSA is misused and insecure; or with knowledge of the private key.

With knowledge of the private key, we simply attempt RSA/OAEP decryption. If it succeeds, we bet for RSA/OAEP (and that will hold except for a vanishingly small proportion of possible plaintext); otherwise we know that's textbook RSA (with certainty, unless there's another option).

With knowledge of the private key, we can still succeed by exploiting weaknesses of textbook RSA; for example:


If we can obtain ciphertexts for the same unknown plaintext enciphered twice, we can compare these ciphertexts. If they match, we bet for textbook RSA (and that will hold with overwhelming probability, unless the random number generator of RSA-OAEP is badly broken). If they differ, we know that's RSA/OAEP.
With knowledge of the public key (assumed in this and the next two bullets), if we know / suppose that the plaintext is a name on the class roll, we try to encipher all names on the class roll (that requires the public key, but by assumption that's known to all), and compare each result to the actual ciphertext. If any match is found, we know with near certainty that's textbook RSA; otherwise, we bet for RSA/OAEP (with odds depending on how confident we are that the plaintext is a name on the class roll).
As suggested in comment: if the ciphertext is an exact power of the public exponent $e$, we know for certain that's textbook RSA. Otherwise, we bet for RSA/OAEP (with odds depending on how likely it is that the plaintext $M$ is small enough that $M^e&lt;N$).
If we can obtain actual ciphertext for a single chosen plaintext, that allows a test.


Also, if we know or can find the size limit for the plaintext, that's also revealing: Textbook RSA usually can encipher messages up to $\lfloor\log_2(N)\rfloor$ bits or $\lfloor\log_2(N)/8\rfloor$ octets; while RSA-OEAP has a much lower limit, $\lceil\log_2(N)/8\rceil-2-2h$ octets, where $h$ is the hash size in octets ($h=32$ for SHA-256).