I have encountered a question asked by my friend, which is stated as follows:
Suppose there exists integers a, b, c, k, l, m such that $ka+lb+mc=1$, then, is the following statement true?
"There exists at least one pair from $(a, b)$, $(b,c)$,$(a,c)$ that is coprime."
Personally, I think the statement is plausible as I can assume one of k, l, and m is zero, which would mean that there exists one pair of coprime numbers. However, I have difficulty in proving it strictly. Many thanks for your help.