Let $n$ and $k$ be integers, $1<k \leq n$. Find an integer $b$ and a set $A$ of $n$ integers satisfying the following conditions:

(i) No product of $k-1$ distinct elements of $A$ is divisible by $b$.

(ii) Every product of $k$ distinct elements of $A$ is divisible by $b$.

(iii) For all distinct $a, a^{\prime}$ in $A$, $a$ does not divide $a^{\prime}$.
Let $p_{1}, \ldots, p_{n}$ be the first $n$ odd primes. Then we can take $A=\left\{2 p_{1}, 2 p_{2}, \ldots, 2 p_{n}\right\}$ and $b=2^{k}$. It is easily seen that the conditions are satisfied.