Determining message given cipher-text and public key
This is called the common modulus attack.

Bezout's Identity says that there exists $x$ and $y$ such that $ax + by = gcd(a, b)$. 

In our case we have $gcd(e_a,e_b) = 1$, so we can find $x$ and $y$ such that $e_{a}x + e_{b}y = 1$ (you can use the extended euclidean algorithm for this).

After solving for $x$ and $y$, you compute:

$C^xC^y\mod N$ to get $M$.

This works because $C^xC^y= M^{e_{a}x}M^{{e_b}y} = M^{e_{a}x + e_{b}y} = M$