We call a positive integer
amazing if there exist positive integers
such that the equality
holds. Prove that there exist
consecutive positive integers which are amazing.
Note. By
we denote the greatest common divisor of positive integers
and
.
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We call a positive integer $n$ amazing if there exist positive integers $a, b, c$ such that the equality $$n = (b, c)(a, bc) + (c, a)(b, ca) + (a, b)(c, ab)$$ holds. Prove that there exist $2011$ consecutive positive integers which are amazing.
Note. By $(m, n)$ we denote the greatest common divisor of positive integers $m$ and $n$.
Školjka