1. ## uniquness proof

Hello everyone, im having a little trouble on this problem...

"Show that greatest common divisor of two integers, which are not both
zero, is unique."

i came across this website: Greatest Common Divisors

any help would be aprreciated

thanks!

2. Originally Posted by matt23721
Hello everyone, im having a little trouble on this problem...

"Show that greatest common divisor of two integers, which are not both
zero, is unique."
That's a weird question: a "greatest" element of anything is always unique. It should seem obvious to you when you think of the set of the common divisors: this is a finite set of integers, it must have a greatest element, and I can't see how one may wonder if it is unique. Here's a proof anyway.
Suppose $a$ is a greatest element of a nonempty set $A$ (here, $A$ is the set of common divisors of two non-zero integers, it is nonempty since it contains 1). Then if $b$ is another greatest element, we would have both $a\geq b$ (because $a$ is larger than any element of $A$) and $b\geq a$ (because $b$ is larger than any element of $A$), hence $a=b$.

3. The gcd of two integers, $x$ and $y$, is an integer $g$ satisfying

1. $g>0$
2. $g | x$ and $g | y$
3. If $z | x$and $z | y$ then $z | g$

While I'm not disagreeing with Laurent's proof, the property of 'greatest' can only be inferred from these.

As a more specific proof of the uniqueness of such an integer, let $g$ and $\tilde{g}$ both satisfy properites 1, 2 and 3 above.

Then property 3 $\Rightarrow ~ g|\tilde{g}$ , $\tilde{g}|g ~ \Rightarrow ~ g = \pm \tilde{g}$

Property 1 $\Rightarrow ~ g,\tilde{g} > 0 ~ \Rightarrow ~ g = \tilde{g}$

4. Originally Posted by Laurent
a "greatest" element of anything is always unique.
Not true. In a ring (for example a unique factorization domain) in which you define a gcd (by the two last propierties pomp gave and that it be nonzero or a unit) this element is not unique since if we multiply it by a unit the new element is also a gcd, and there may not be an order that let's you decide which is 'the one'.

5. thanks... starting to make some sense... how would i formulate that into a proof though? I sort of understand it in words... just writing it out in "math language" is a whole other story

6. Sorry if they didn't help, but both mine and Laurent's proofs were 'mathematical'. Was there something more you were looking for?

7. sorry i did not meant to sound rude! what i guess i meant to say is how to a go about showing its uniqueness

8. Originally Posted by matt23721
sorry i did not meant to sound rude! what i guess i meant to say is how to a go about showing its uniqueness
No worries.
Usualy in mathematics, if you're asked to prove something is unique, you assume it isn't. Then, you show that assuming this, you arrive at a contradiction.
This is what I did in my post when I assumed that $g$ and $\tilde{g}$ were both gcd's.

9. oh ok .. i follow you... instead of using g and g- could i use say one is g and the other "g" is say f??

10. Originally Posted by matt23721
oh ok .. i follow you... instead of using g and g- could i use say one is g and the other "g" is say f??
Yeh. You can call it shilrley if you want

11. haha thanks mate.. i appreciate it tons.. message me ur aim sn or facebook name if ud like, i might have a few questions later if u wouldnt mind helpn

12. hey pomp... real quick, whatd u mean by z where did that come from?

13. Originally Posted by matt23721
hey pomp... real quick, whatd u mean by z where did that come from?
z is just any old integer. So property 3 is just saying that if there is any old integer z, such that z divides both x and y, the z must also divide the gcd of x and y.

14. thanks man! i understand.... i sent u a message on fb