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# Help Please! Let $x$ and $y$ be integers. Show that $9x + 5y$ is divisible by 19 if and only if $x + 9y$ is divisible by 19.

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Let $x$ and $y$ be integers. Show that $9x + 5y$ is divisible by 19 if and only if $x + 9y$ is divisible by 19.

This is a repost, though the last post's solution was removed for some reason.

Thanks!

Aug 3, 2020
edited by new3r  Aug 3, 2020

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$$\text{Let x and y be integers. Show that 9x + 5y is divisible by 19 \\if and only if x + 9y is divisible by 19.}$$

My attempt:

$$\begin{array}{|rcll|} \hline x + 9y &\equiv& 0 \pmod{19} \\ x + 9y &=& 19n,\ \qquad n\in \mathbb{Z} \\ \mathbf{x} &=& \mathbf{19n -9y} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline 9x + 5y &\overset{\color{red}?}{\equiv}& 0 \pmod{19} \quad | \quad \mathbf{x=19n -9y} \\ 9(19n -9y) + 5y &\overset{\color{red}?}{\equiv}& 0 \pmod{19} \\ 9*19n -81y + 5y &\overset{\color{red}?}{\equiv}& 0 \pmod{19} \\ 9*19n -76y &\overset{\color{red}?}{\equiv}& 0 \pmod{19} \\ 9*{\color{red}19}n -4*{\color{red}19}y & \equiv& 0 \pmod{19} \checkmark \\ \hline \end{array}$$

Aug 3, 2020