Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

If $x$,$y$ are odd integers, argue that

$$\gcd(x,y)=\gcd\left(\frac{x+y}{2},\frac{x-y}{2}\right)\;.$$

I'm having a difficult time with this:

First, I tried a few examples to check that my professor didn't once again make an obvious mistake, because he's very good at that. Poor guy..

$$\gcd(5,3)=\gcd\left(\frac{5+3}{2},\frac{5-3}{2}\right)=1$$

$$\gcd(15,3)=\gcd\left(\frac{15+3}{2},\frac{15-3}{2}\right)=3$$

share|improve this question

1 Answer 1

up vote 2 down vote accepted

HINT: Suppose that $d\mid x$ and $d\mid y$; then clearly $d\mid x+y$ and $d\mid x-y$. Since $x$ and $y$ are odd, $x+y$ and $x-y$ are even, so there are integers $m$ and $n$ such that $x+y=2m$ and $x-y=2n$. Thus, $d\mid 2m$ and $d\mid 2n$. But $d$ must be odd (why?), so $d\mid m$ and $d\mid n$. Thus, $\gcd\{x,y\}\mid\gcd\{m,n\}$.

Now suppose that $d\mid m$ and $d\mid n$. Can you show that $d\mid x$ and $d\mid y$, so that you can conclude that $\gcd\{m,n\}\mid\gcd\{x,y\}$?

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.