Help with Simple Cracking Challenge

Hello,

For one of my subjects at university we have been given an assignment in which we had to implement two simple encryption algorithms. These are the Caesar Cipher and the XOR Cipher. I've had no trouble implementing them, however we have been given two "Mystery" files in which they have been encrypted using one of these ciphers and the challenge is to attempt to decrypt the file without knowing the algorithm used or the key.

I should probably mention how the algorithms work. The Caesar Cipher is slightly modified so that a message is considered as a series of bytes rather than letters. The shift length is the key for this algorithm.

The XOR Cipher generates a keystream from a pseudo-random number generator (PRNG) which is the length of the message. The XOR operation is then applied:
ciphertext[i] = plaintext[i] ^ keystream[i];
The key for this algorithm is the seed to the PRNG. The PRNG used is not cryptographically secure, it is a simple linear congruential generator that always generates the same sequence.

I have managed to successfully decrypt the first Mystery file, it was encrypted using the XOR Cipher and the plaintext was simply an ASCII text file. My program simply went through all possible keys for each cipher, decrypted the file and then checked to see if the decrypted result contained valid ASCII bytes.

I'm having more trouble with the second file. We have been told that the plaintext is not a text file but a file of another well known type. I'm pretty sure that the file is encrypted using the XOR Cipher as I have gone through all 255 possible keys for the Caesar Cipher and couldn't find anything.

So far, I have tried to decrypt the file by going through all different keys, decrypting the file and checking the bytes at the start of the file for identifiers that are present in order to tell the file type. I haven't had much luck with this, and I'm sure there's probably a much simpler way to break this, but I can't seem to figure it out.

Is there some sort of property of XOR that could be used to help decrypt this file? Or something to do with PRNG? I feel kind of stupid that I can't figure it out, apparently someone else cracked it the same day that it was released .

Any hints would be much appreciated. I'd prefer hints than full solutions as I'd like to figure this out myself if I can. Oh, the file is 328 bytes long which seems rather small, I'm not really sure what type of file it could be.

NOTE: This isn't for marks, it's just a little extra thing we have been given. I just want to figure it out, it's been driving me crazy.

Hopefully I've given enough information.

Thanks!

UPDATE: I've managed to decrypt the file, thanks for the help . I basically went through all possible keys, decrypted the file and checked to see if the file type was either an image file or zip file. It turned out to be a zip file.