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I'm having some trouble understanding a proof on Naoki Sato's notes on Number Theory and I was wondering if you guys could give me some help. The problem is that I don't understand the last implication on the proof for example 1.1

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. $17\ |\ (2x + 3y) \implies 17\ |\ [13(2x + 3y)]$, or $17\ | \ (26x + 39y) \implies 17\ |\ (9x + 5y)$ conversely, $17\ |\ (9x + 5y) \implies 17 |\ 4(9x + 5y)$, or $17\ |\ (36x + 20y) \implies 17\ |\ (2x + 3y)$.

My problem is that I don't understand how does $17\ |\ (26x + 39y) \implies 17\ |\ (9x + 5y)$. If you could elaborate on this step I would be most grateful.

I'm sorry if this is an obvious question but I am a beginner and I just can't get it.

Thanks for your help in advance.

Bill Dubuque
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    Hello and welcome to Math.SE! This is a pretty well-written first question. I've changed your title to be more immediately informative to other users at a glance. You might notice that it's formatted nicely --- we use mathjax to have beautiful math typography. I hope you continue to ask good questions (or perhaps answer them) in the future. – davidlowryduda Oct 14 '16 at 20:51

2 Answers2

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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 $p\mid (a+b)$.

Dietrich Burde
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One has $9x + 5y = (26x + 39y) - 17(x+2y)$, and $17$ clearly divides the right hand side if $17 \mid 26x + 39y$.

arkeet
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