Given one message encrypted repeatedly with the same 'n' but different 'e's, is it possible to decrypt the ciphertext without simply bruteforcing n?
Ok, here's a hint:

Suppose you know the values of $P^{e_1} \bmod N$ and $P^{e_2} \bmod N$.  How could you use those two values to compute $P^{e_1 - e_2} \bmod N$?  How could you then use this repeatedly to compute $P^{\gcd(e_1, e_2)} \bmod N$?