linear congruence

The linear congruence

$$ax\equiv b(modm),$$

where $a$, $b$ and $m$ are known integers and  $\gcd (a,m)=1$,  has exactly one solution $x$ in $\mathbb{Z}$, when numbers congruent to each other are not regarded as different.  The solution can be obtained as

$$x=a^{\phi (m)-1}b,$$

where $\phi$ means Euler’s phi-function.

Solving the linear congruence also gives the solution of the Diophantine equation

$$ax-my=b,$$

and conversely.  If  $x=x_0$,  $y=y_0$  is a solution of this equation, then the general solution is

$$\{\begin{array}{cc}x=x_0+km,\hfill & \\ y=y_0+ka,\hfill & \end{array}$$

where  $k=0$, $\pm 1$, $\pm 2$, …