Check out [the detailed writeup](https://rootfs.eu/codeblue2017-common-modulus/)

We're given two RSA-encrypted ciphertexts of the same message (flag), under different keys that share the same modulus.

Calculate $c_1^{x} * c_2^{y} \mod n$ such that $x$ and $y$ are integers that satisfy Bezout's identity for the public exponents $e_1$ and $e_2$ respectively. Use the Extended Euclidean Algorithm to get $x$ and $y$.