If 1 can be a linear integer combination of 3 non zero integers then Is at least one of the three possible pairs coprime?

Question

If a,b and c are non zero integers and that 1 can be written as a linear integer combination of them such that k, m and n are integers and:

Ka + mb +nc = 1 then surely by bezouts identity, atleast one of the pairs (a,b),(a,c) and (b,c) must be co prime?

I’ve tried dissecting this but I can’t seem to get to any solid proof that this could be correct?

And light shed on this would be great

Answer 1

$6+10-15=1$ and that's it but answer needs 30 characters.