1. A positive integer is called sparkly if it has exactly 9 digits, and for any $n$ between 1 and 9 (inclusive), the $n^{\text {th }}$ digit is a positive multiple of $n$. How many positive integers are sparkly?
Solution: For each $n=1,2, \ldots, 9$ there are $\lfloor 9 / n\rfloor$ different possibilities for the $n^{\text {th }}$ digit. For example there are $\lfloor 9 / 2\rfloor=4$ possible choices for the second digit (these being 2, 4, 6 and 8). Therefore the answer is
$$
\left\lfloor\frac{9}{1}\right\rfloor \times\left\lfloor\frac{9}{2}\right\rfloor \times\left\lfloor\frac{9}{3}\right\rfloor \times\left\lfloor\frac{9}{4}\right\rfloor \times\left\lfloor\frac{9}{5}\right\rfloor \times\left\lfloor\frac{9}{6}\right\rfloor \times\left\lfloor\frac{9}{7}\right\rfloor \times\left\lfloor\frac{9}{8}\right\rfloor \times\left\lfloor\frac{9}{9}\right\rfloor
$$
which equals $9 \times 4 \times 3 \times 2 \times 1 \times 1 \times 1 \times 1 \times 1=216$.