Can you help me understand the RSA Broadcast Attack?
Your solution works for Part 1 because (there) for each $i \neq j$ holds  $gcd(N_i,N_j) = 1$. In Part 2 $gcd(N_7, N_{13})$ is not equal to $1$.
$N_{13}$ is a product of two primes ($q = gcd(N_7, N_{13})$ and $p = N_{13}/q$).
Using that information you can recover the secret key $d$ (simply invert $e$ modulo $(p-1)(q-1)$ as during the key generation of RSA) and from that you can recover $m = C_{13}^d (mod\ N_{13})$. We did not use the broadcast attack in this case (I am not sure if you can make it work).