Smallest integer with digit product ten thousand

Find the least positive integer whose digits have product $10000$.

$255558$. Since $10000=2^4\cdot5^4$, we need four $5$s, and the $2^4$ is packed into as few digits as possible: two digits ($2,8$ or $4,4$). Six digits is the minimum, and arranging $2,5,5,5,5,8$ in increasing order gives $255558$, smaller than $445555$.