Finding number of solutions to a congruence equation

Question

Let the congruence equation be:

$$ 51813x \equiv 14593 \pmod{119472}$$

How can you determine the number of roots that exist for this equation? Is there also a way to find the smallest/largest roots? I am under the impression I should use the chinese remainder theorem but I am not sure where. Any help is appreciated.

Answer 1

Hint: There are no solutions for $x$. Compute the gcd(51813,119472).

Answer 2

The equation $ax \equiv b \bmod m$ is equivalent to $ax=b+my$.

Therefore, $ax \equiv b \bmod m$ has solutions iff $\gcd(a,m)$ divides $b$.

Try that for your equation.