Understanding a proof that $2x + 3y$ is divisible by $17$ iff $9x + 5y$ is divisible by $17$

proof-explanation elementary-number-theory divisibility number-theory

Solution 1:

If $17\mid (26x+39y)$, and $17\mid (-17x-34y)$, then we may add to get $17\mid 9x+5y$. In general the rule is, if $p\mid a$ and $p\mid b$, then $p\mid (a+b)$.

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