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I am currently reading a book on basic number theory and in the first chapter the author uses the fact that if $d \mid x$ and also $d\mid y$, then it is true that $d$ is a divisor to both sides of the equation: $$ x = y +r $$

I see that it is true, but how could you prove that? I mean, if $x$ and $y$ are even, then the $\gcd(x,y)$ is even and the rest $r = 0$, hence for any other divisor $c < \gcd(x,y)$ the rest $r$ will of course be even.

What about if $\gcd(x,y)$ is odd? Or is this a bad way to approach a proof for above statement? How would you do to prove it?

Best regards,

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If $d$ divides $x,y$ ,let $\frac xX=\frac yY=d\implies r=x-y=d(X-Y)$ –  lab bhattacharjee May 28 '13 at 18:10
    
By assumption, $d$ divides the left side of the equation, $x$. Since the two sides are equal, $d$ also divides the right side. –  Erick Wong May 28 '13 at 18:51

1 Answer 1

up vote 3 down vote accepted

We are given that $d\mid x$ and $d\mid y$. So there are integers $m, n$ such that $$\;\;x = md, \quad \text{and}\quad y = nd$$

Then given $x = y + r \iff x - y = r$, we have $$x - y = md - nd = (m-n)d = r.$$

Hence $d\mid r$, as well, since $m, n$ are integers.

Then what can you say about $d$ with respect to the right-hand side: $$y + r = nd + (m-n)d = (n+m-n)d\quad?$$

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Nice! Now I see it clearly, thanks! –  Math83 May 28 '13 at 19:08
    
You're welcome! –  amWhy May 28 '13 at 19:08
    
@amWhy: Great feedback! =1 –  Amzoti May 29 '13 at 0:40
    
It was a pretty good day for me, feedback-wise! That's the most rewarding part of answering questions ;-) –  amWhy May 29 '13 at 0:41

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