Math Problem Solver

There are multiple solutions for $x$ in the congruence relation $3 x \equiv 9 \pmod{18}$: $\textcolor{#228B22}{x \in \left\{ 3, 9, 15\right\} }$

Remove the greatest common factor shared between the left-hand side, the right-hand side, and the modulus. Dividing each by $3$ yields $x \equiv 3 \pmod{6}$. From this, we know that $3 x \equiv 9 \pmod{18}$ has $3$ solutions.

This is a different expression in a different modulus, but can be used to compute the solutions.

We were originally solving $3 x \equiv 9 \pmod{18}$. In mod $18$, there are $3$ numbers that are solutions. Given that there is one unique solution in mod $6$, the other solutions are equivalent in mod $6$, but distinct in mod $18$. These distinct solutions can be produced by adding the modulus $6$ to the result until the result is no longer less than $18$, meaning the solutions to $3 x \equiv 9 \pmod{18}$ are $3, 9, 15$.

These solutions can be verified by plugging them back into the original reduced equation $3 x \equiv 9 \pmod{18}$: $$\begin{align*} & 3 \cdot 3 & = \; & 9 & \equiv \; & 9 \pmod{ 18 } \\ & 3 \cdot 9 & = \; & 27 & \equiv \; & 9 \pmod{ 18 } \\ & 3 \cdot 15 & = \; & 45 & \equiv \; & 9 \pmod{ 18 } \\ \end{align*}$$