Greatest common divisor

Let $gcd(x,y)=k.$ Let $x=km,y=kn$ where $gcd(m,n)=1$ . Now the given equation reduces to $km \cdot kn \cdot k=1080 \implies k^3 mn=3^3 2^3 5.$

To solve these cases, it's very important to recall that $gcd(m,n)=1$ AND that we have to search for unordered pairs.

Possible cases to consider :

1) $k=1 \implies mn=27×8×5 \implies m(,n)={{(1080,1); (135, 8); (216 ,5); (27, 40)} }$

2) $k=2 \implies mn=135 \implies (m,n)={(135,1); (27,5) }$

3) $k=3 \implies mn=40 \implies (m,n)={(40,1); (8,5)}$ .

4) $k=6 \implies mn=5 \implies (m,n)={(5,1)}$

Thus total $\boxed{9}$ pairs.