12. (9th grade) Find all increasing arithmetic progressions of natural numbers, in which for any natural $n$ the product of the first $n$ terms divides the product of the next $n$ terms (from $n+1$ to $2n$).
Answer: $k, 2 k, 3 k, \ldots, k \in \mathbb{N}$.

Solution. 1. Divide the given progression by the GCD of all its terms and obtain a new progression that still satisfies the condition, say, $a, a+d, \ldots$, where $(a, d)=1$ and $d \geq 1$. Since by the condition (for $n=1$) $a \mid a+d$, then $a \mid d$ and $a=1$.