HMMT-Nov 2010 guts, Problem 32

Let $T$ be the set of numbers of the form $2^{a} 3^{b}$ where $a$ and $b$ are integers satisfying $0 \leq a, b \leq 5$. How many subsets $S$ of $T$ have the property that if $n$ is in $S$ then all positive integer divisors of $n$ are in $S$ ?

Solution: Consider the correspondence $(a, b) \leftrightarrow 2^{a} 3^{b}$ for non-negative integers $a$ and $b$. So we can view $T$ as the square of lattice points $(a, b)$ where $0 \leq a, b \leq 5$, and subsets of $T$ as subsets of this square. Notice then that the integer corresponding to $\left(a_{1}, b_{1}\right)$ is a divisor of the integer corresponding to $\left(a_{2}, b_{2}\right)$ if and only if $0 \leq a_{1} \leq a_{1}$ and $0 \leq b_{1} \leq b_{2}$. This means that subsets $S \subset T$ with the desired property,\\ correspond to subsets of the square where if a point is in the set, then so are all points to the left and south of it. Consider any such subset $S$. For each $0 \leq x \leq 5$, let $S_{x}$ be the maximum $y$ value of any point $(x, y) \in S$, or -1 if there is no such point. We claim the values $S_{x}$ uniquely characterize $S$. This is because each $S_{x}$ characterizes the points of the form $(x, y)$ in $S$. In particular, $(x, z)$ will be in $S$ if and only if $z \leq S_{x}$. If $(x, z) \in S$ with $z>S_{x}$, then $S_{x}$ is not the maximum value, and if $(x, z) \notin S$ with $z \leq S_{x}$, then $S$ fails to satisfy the desired property. We now claim that $S_{x} \geq S_{y}$ for $x<y$, so the sequence $S_{0}, \ldots, S_{1}$ is decreasing. This is because if $\left(y, S_{y}\right)$ is in the set $S$, then so must be $\left(x, S_{y}\right)$. Conversely, it is easy to see that if $S_{0}, \ldots, S_{1}$ is decreasing, then $S$ is a set satisfying the desired property. We now claim that decreasing sequences $S_{0}, \ldots, S_{5}$ are in bijective correspondence with walks going only right and down from $(-1,5)$ to $(5,-1)$. The sequence $S_{0}, \ldots, S_{5}$ simply corresponds to the walk $(-1,5) \rightarrow\left(-1, S_{0}\right) \rightarrow\left(0, S_{0}\right) \rightarrow\left(0, S_{1}\right) \rightarrow\left(1, S_{1}\right) \rightarrow \cdots \rightarrow\left(4, S_{5}\right) \rightarrow\left(5, S_{5}\right) \rightarrow(5,-1)$. Geometrically, we are tracing out the outline of the set $S$. The number of such walks is simply $\left(\begin{array}{c}12 \\ 6\end{array}\right)$, since we can view it as choosing the 6 of 12 steps at which to move right. Thus the number of subsets $S$ of $T$ with the desired property is $\left(\begin{array}{c}12 \\ 6\end{array}\right)=924$. $\fbox{924}$.

Answer: 924