Sam and Tim have set up their RSA keys $(e_s, n), (e_t, n),$ respectively, where the n-value is the same. Furthermore, it happens that $\gcd (e_s, e_t) = 1$. Suppose that their friend Rob wants to send both Sam and Rob a message M that is corprime with n. Rob encrypts M using Sam and Tim's public keys producing ciphertext $C_s$ and $C_t$ respectively.
Prove that anyone who obtains the values of both $C_s$ and $C_t$ can determine the message M using the public keys of Sam and Tim, without knowing their private keys.
So far, I've used the fact that $\gcd(e_s, e_t) = 1$ to get $ae_s + be_t = 1.$ From there, I said that $C_s^a C_t^b = M^{ae_s + be_t} \bmod n$. Since $ae_s + be_t = 1$, then $C_s^a C_t^b = M \bmod n$. As such, anyone who knows $C_s, C_t$ and the public keys can find the value of M. Is this solution correct?
Your solution is basically correct. The argumenent is: get the Bezout coefficients $a,b\,$ with $a e_s + b e_t =\gcd(e_s,e_t)=1\;$ and compute $$M=M^1= M^{a e_s + b e_t} = M^{a e_s} M^{ b e_t} = (M^{e_s})^a (M^{e_t})^b \equiv C_s^a C_t^b \mod n$$