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How to prove that a result does not hold in general

1. The problem statement, all variables and given/known data
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.

2. Relevant equations


3. 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.
 

phyzguy

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It only takes a single counter-example to prove the second part.
 

fresh_42

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Looks good so far. You can work with ##(a,b)=3## as an example. Just find specific numbers ##a,b,c## so that the statement is false.
 
Looks good so far. You can work with ##(a,b)=3## as an example. Just find specific numbers ##a,b,c## so that the statement is false.
Thank you for the reply, I did find values for a, b, and c.
 

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