- #1

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## Homework Statement

Prove that for all a, b, c ∈ ℤ, if a|c and b|c and (a,b)=1 then ab|c. Prove result does not hold in general when (a,b)≠1.

## Homework Equations

## The Attempt at a Solution

This is not my formal proof it's just the scratch work, for the first part, I have there are integers x and y such that ax + by = 1 which leads us to cax + cby = c. Since a|c and b|c there are integers f and g such that c=fa and c=gb which leads us to gbax + faby = c. Since ab divides the left side it must mean it divides the right side. I am just confused how to prove the second part of the question. I thought instead I could do (a,b)=3 and show that it does not work but I do not think that is right. Any suggestions would be appreciated.

Thank you.