Congruences

TL;DR: The congruence \(ax\equiv b(mod \ m)\) is equivalent to the equation \(ax-my=b\). If \(c\) does not divide \(b\), then the equation has no solutions. If \(c\mid b\), then there are infinitely many solutions. The number of incongruent solutions is given by the set \[x=x_0+(m/c)t\), where \(t\) is taken modulo \(c\).

Abstract: As we mentioned earlier, \(ax\equiv b(mod \ m)\) is equivalent to \(ax-my=b\). By Theorem 19 on Diophantine equations, we know that if \(c\) does not divide \(b\), then the equation, \(ax-my=b\) has no solutions. Notice also that if \(c\mid b\), then there are infinitely many solutions whose variable \(x\) is given by \[x=x_0+(m/c)t\] Thus the above values of \(x\) are solutions of the congruence \(ax\equiv b(mod \ m)\). Now we have to determine the number of incongruent solutions that we have. Suppose that two solutions are congruent, i.e. \[x_0+(m/c)t_1\equiv x_0+(m/c)t_2(mod \ m).\] Thus we get \[(m/c)t_1\ equiv (m/c)t_2(mod \ m).\] Now notice that \((m,m/c)=m/c\) and thus \[t_1\equiv t_2(mod \ c).\] Thus we get a set of incongruent solutions given by \(x=x_0+(m/c)t\), where \(t\) is taken modulo \(c\).