Question: Find integers $ a,b, $ and $ c $ where $ \gcd(a,b,c) = 1 $, but $ \gcd(a,b) \neq 1 $, $ \gcd(a,c) \neq 1 $, and $ \gcd(b,c) \neq 1 $. 

I tried so many combinations but I can't find 3 integers that meet these requirements.

I even though $ (0,0,0) $ works, because I tried to convince myself 1 is the first positive integer where 0 has a divisor, because you can't divide by 0. I am not sure if there is a more systematic approach to this.

Answer: 
Consider $a = 6, b= 10, c = 15$

An easy way to construct these is by considering three prime $2, 3, 5$, then pairwise multiply them.
Choose any three prime numbers $p,q,r$ and create the products $pq,qr,pr$. Each product lacks one of the prime factors, so the largest number that divides all three is $1$, but every pair of products has a factor in common, and hence a gcd greater than $1$. This is a generalization of the rather specific answer given first.