Suppose in two RSA instances the same $p,q,N$ are used, but different public keys $a,b$ (and corresponding private keys)
Suppose now we have the two equations
$c_{1}=m^{a} \bmod N$
$c_{2}=m^{b} \bmod N$
Is it possibe to retrieve the original message $m$ with this information?
It is possible to retrieve $m$ if $a$ and $b$ are coprime.
This means $GCD(a,b)=1$. Then, applying the Extended Euclidiean Algorithm we have $$ax+by=GCD(a,b) = 1$$ Therefore, $$c_1^x.c_2^y = m^{ax}.m^{by} = m^{ax+by} = m\bmod N$$
So just like that, we could recover $m$ but this gives us no information about the factorization of $N=pq$.
Is it possible to retrieve the original message $m$ with this information?
Well, this is a standard exercise for beginners (that is, you're supposed to learn from it), and so I won't spell out the answer.
I will give you a hint: if you know $N, c_1 = m^a \bmod N, c_2 = m^b \bmod N$, can you compute the value of $c_3 = m^{a-b} \bmod N$? If so, how could you exploit that?
And, at the end, what condition must exist between $a$ and $b$ to allow you to recover $m$?
