Question 5
The number $$
\small \sqrt{104\sqrt{6} + 468\sqrt{10} + 144\sqrt{15} + 2006}
$$ can be written as $a\sqrt{2} + b\sqrt{3} + c\sqrt{5},$ where $a,$ $b,$ and $c$ are positive integers. Find $a \cdot b \cdot c.$
\small \sqrt{104\sqrt{6} + 468\sqrt{10} + 144\sqrt{15} + 2006}
$$ can be written as $a\sqrt{2} + b\sqrt{3} + c\sqrt{5},$ where $a,$ $b,$ and $c$ are positive integers. Find $a \cdot b \cdot c.$