I was studying the number theory and came across this question.
Example 1.1. Let $x$ and $y$ be integers. Prove that $2x + 3y$ is divisible by 17 if and only if $9x + 5y$ is divisible by 17.
Solution. $17 \mid (2x + 3y) \implies 17 | [13(2x + 3y)]$, or $17 \mid (26x + 39y) \implies 17 \mid (9x + 5y)$. Conversely, $17 \mid (9x + 5y) \implies 17 \mid [4(9x + 5y)]$, or $17 \mid (36x + 20y) \implies 17 \mid (2x + 3y)$.
I have a difficulty understanding how $$17\mid(26x+39y)$$ implies $$17\mid (9x+5y)$$ and vice versa.
I know this question as already been asked here but I didn't understand from that answer and since I'm new to Math SE, I don't have enough points to add a comment to that post to clarify that answer.
Best Answer
From $26x+39y$ you can subtract any multiple of $17$, so you can substract $17x+34y$ to get $9x+5y$.