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Suppose $x_1, x_2, y_1, y_2$ are known integers which satisfy

$y_1 \equiv \gamma x_1 \pmod c$


$y_2 \equiv \gamma x_2 \pmod c$

where $\gamma$ and $c$ are unknown integers. Is there a way to determine the values of $\gamma$ and $c$?

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There's at least a trivial set of solutions: c = 1 or -1, any gamma. – Weltschmerz Sep 30 '10 at 13:27
up vote 3 down vote accepted

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.

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