Question 1
Let $M \ge 1$ be a real number. Determine all natural numbers $n$ for which there exist pairwise distinct natural numbers $a, b, c \gt M$, such that
$$
n = (a, b) \cdot (b, c) + (b, c) \cdot (c, a) + (c, a) \cdot (a, b)
$$
(where $(x, y)$ denotes the greatest common divisor of natural numbers $x$ and $y$).
$$
n = (a, b) \cdot (b, c) + (b, c) \cdot (c, a) + (c, a) \cdot (a, b)
$$
(where $(x, y)$ denotes the greatest common divisor of natural numbers $x$ and $y$).