Together, the products by four and by five of an integer use each digit from 1 to 9 once, and only once. What is this number?
Let $n$ be the number such that $4n$ and $5n$ are written with the 9 digits. $5n \geq 4n$ so $5n$ has at least 5 digits and $4n$ at most 4. Thus, $4n \leq 9876 \leq 12345 \leq 5n$. Now $9876 = \frac{4}{5} \times 12345$, so $4n = 9876$ and $5n = 12345$, thus $n = 2469$.