MHB How can I prove that gcd(a,b) is an integer and gcd(a/b, b/h) = 1?

[simo1]

can I get hints on how to prove this question:

let a,b be in Z(integer) with h=gcd(a,b) not equal to zero then [(a divide b), (b divide h)] are in Z. and gcd[ (a divide h, b divide h)]=1

 

[Nono713]

Did you mean "a divide h"? The first part is pretty much the definition of the greatest common divisor ("the greatest integer which divides both numbers", so trivially $h$ divides both $a$ and $b$). What definition of the gcd are you working from? For the second part, suppose that $a/h$ and $b/h$ shared a common factor, so that their gcd was not equal to $1$. Would that not conflict with the definition of $h = \gcd(a, b)$? Can you see the problem?

 

[DavidCampen]

Let me restate the problem to see if I understand it correctly.


Given that a,b are integers, h=gcd(a,b) and h not equal to 0 then
is it true that:

1. a divided by b and b divided by h are integers
2. gcd(a divided by h, b divided by h)=1

Statement 2 is true and is given as Theorem 1 in "Elementary Number Theory" by Dudley.

In Statement 1, there is no reason why a divided by b should be an integer.