Finding three numbers that are pairwise not relatively prime, but with $gcd(a,b,c)=1$

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.