They grew up so beautifully

Find the number of solutions to $[a,b,c](a,b,c) = \sqrt{abc}$ where $a\geq b \geq c$ are integers ranging from $1$ to $30$ .

Here $[a,b,c]$ and $(a,b,c)$ are the least common multiple (LCM) and the greatest common divisor (GCD) of $a,b,c$ respectively.