49th Mathematical Olympiad in Ukraine

Since $5120=2^{10}\cdot 5$, then the main question is – how many digits have its number? Obviously among these digits it hasn't $0$ and has $5$ because only $5$ from these digits divides by $5$. From the ten factors $2$ we can produce at the least four digits – $2$, $8$, $8$, $8$ or $4$, $4$, $8$, $8$. Therefore from two sets of digits $5$, $2$, $8$, $8$, $8$ and $5$, $4$, $4$, $8$, $8$ we must produce a minimum of numbers. The minimum of first figure is $2$ so that the required minimum integer number is $25888$.