ILL 1974 Problem 28

Let $M$ be a finite set and $P=\{ M_1,M_2,\ldots ,M_l\}$ a partition of $M$ (i.e., $\bigcup_{i=1}^k M_i, M_i\not=\emptyset, M_i\cap M_j =\emptyset$ for all $i,j\in\{1,2, \ldots ,k\} ,i\not= j)$ . We define the following elementary operation on $P$ : Choose $i,j\in\{1,2,\ldots ,k\}$ , such that $i=j$ and $M_i$ has a elements and $M_j$ has $b$ elements such that $a\ge b$ . Then take $b$ elements from $M_i$ and place them into $M_j$ , i.e., $M_j$ becomes the union of itself and a $b$ -element subset of $M_i$ , while the same subset is subtracted from $M_i$ (if $a=b$ , $M_i$ is thus removed from the partition). Let a finite set $M$ be given. Prove that the property "for every partition $P$ of $M$ there exists a sequence $P=P_1,P_2,\ldots ,P_r$ such that $P_{i+1}$ is obtained from $P_i$ by an elementary operation and $P_r=\{M\}$ " is equivalent to "the number of elements of $M$ is a power of $2$ ."