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Suppose I have a set of $n$ numbers and their gcd is $d$. If I divide every number by $d$, is it possible to find a pair that is relatively prime? Intuitively yes, but how do I prove it? I tried proceeding by contradiction, like what happens when there are no two numbers who don't share a divisor.

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You can't prove this because it's not true. Consider

$$\{6,10,15\}$$

The $\gcd$ of all of them is $1$, but there is no pair whose $\gcd$ is $1$.

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