Suppose $x_1, x_2, y_1, y_2$ are known integers which satisfy
$y_1 \equiv \gamma x_1 \pmod c$
and
$y_2 \equiv \gamma x_2 \pmod c$
where $\gamma$ and $c$ are unknown integers. Is there a way to determine the values of $\gamma$ and $c$?
Suppose $x_1, x_2, y_1, y_2$ are known integers which satisfy
$y_1 \equiv \gamma x_1 \pmod c$
and
$y_2 \equiv \gamma x_2 \pmod c$
where $\gamma$ and $c$ are unknown integers. Is there a way to determine the values of $\gamma$ and $c$?
No, because if c=pq, for primes p and q, then both congruences hold mod p and mod q as well. So c is not uniquely determined.
For example, if $y_1=y_2=11$ and $x_1=x_2=1$ and $\gamma=5$, c could be 1,2,3 or 6.