Sam and Tim have set up their RSA keys (eS; n); (eT; n), respectively, where the n-value is the same. Furthermore, it happens that gcd(eS;eT) = 1. Suppose that their friend Rob wants to send both Sam and Tim a message M that is coprime with n. Rob encrypts M using Sam and Tim’s public keys, producing the ciphertexts CS and CT respectively.
Prove that anyone who eavesdrops and obtains the values of both CS and CT can determine the message M using the public keys of Sam and Tim, without knowing their private keys.
Hint: Apply Bézout's identity to find $a,b$ such that $a\cdot eS + b \cdot eT=1$ Now use the laws of exponents.
Added: I think you are just supposed to say that such $a,b$ exist. You can't find them specifically without $eS,eT$ If you did know $eS,eT$, you would use the Extended Euclidean algorithm to find them. Then $CS^a\cdot CT^b \equiv (M^{eS})^a\cdot (M^{eT})^b \equiv M^{(a\cdot eS + b \cdot eT)}\equiv M \pmod n$