Crack RSA with additional information Suppose $N=3953$, Alice encryption key is $e_A = 337$ and bobs is $e_B=23$. Eve happens to know the corresponding decryption keys, $d_A=3385$ and $d_B=2663$. Can I use this information to find the prime factorization of $N$?
Here you have an overkill of information, really. We know that $e_A$ and $d_A$ are related by the fact that $e_A d_A \equiv 1 \pmod{\phi(N)}$, normally, and then this answer gives an algorithm.

But here we have more info, as both $e_A d_A - 1$ and $e_B d_B - 1$ are multiples of $\phi(N) = (p-1)(q-1)$. We need some multiple of their gcd, that is close to $N$ to get $\phi(N)$. 
Their gcd is 7656, and half of that is 3828, which is probably $\phi(N) = (p-1)(q-1)$.

Now from this you can solve the quadratic equation $(x-p)(x-q) = 0$, because the left hand side is now known: $x^2 - (p+q)x + pq = x^2 + (N - \phi(N) + 1 )x + N = 0$, and $p$ and $q$ follow from the standard formula.