Question 29
Let $n$ be a positive integer and $M = \{1, 2, \dots, 2n + 1\}$. Find out in how many ways we can split the set $M$ into three mutually disjoint nonempty sets $A, B, C$ so that both the following are true:
(i) for each $a \in A$ and $b \in B$, the remainder of the division of $a$ by $b$ belongs to $C$,
(ii) for each $c \in C$ there exists $a \in A$ and $b \in B$ such that $c$ is the remainder of the division of $a$ by $b$.
Mircea Fianu
(i) for each $a \in A$ and $b \in B$, the remainder of the division of $a$ by $b$ belongs to $C$,
(ii) for each $c \in C$ there exists $a \in A$ and $b \in B$ such that $c$ is the remainder of the division of $a$ by $b$.
Mircea Fianu