[Math] Find all numbers $r$ for which the system of congruences is true

Question

Find all numbers $r$ for which the system of congruences
\begin{align*}
x &\equiv r \pmod{6}, \\
x &\equiv 9 \pmod{20}, \\
x &\equiv 4 \pmod{45}
\end{align*}has a solution.

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Guess and check won't work, thanks in advance for giving a solution!

Answer 1

From the second equation, since $x \equiv 9 \pmod{20}$, we know $x$ is odd. Then $r$ can be one of $1$, $3$, and $5$. From the third equation, $x \equiv 4 \pmod{45}$. We see that $x \equiv 1 \pmod{3}$ (since $x = 45y+4$, for some integer $y$). Then we see that $r$ can only be $1$ (since if $x \equiv 3 \mod{6}$, it will have remainder $0$ when divided by $3$ and if $x \equiv 5 \mod{6}$, it will have remainder $2$ when divided by $3$.