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

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To prove that gcd(a,b) is an integer and that gcd(a/h, b/h) = 1, it is essential to recognize that h, defined as gcd(a,b), divides both a and b, confirming its integer status. The discussion highlights that if a/h and b/h shared a common factor, it would contradict the definition of h as the greatest common divisor. The first statement regarding a divided by b being an integer is disputed, as there is no inherent reason for this to be true. The second statement is affirmed as true and is referenced from "Elementary Number Theory" by Dudley. Overall, the key takeaway is the relationship between the definitions of gcd and the implications of integer division.
simo1
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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
 
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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?
 
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.
 
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