Baltic Way Problem Shortlist, 2011

Answer: $d = 1$, $d = 3$ or $d = 9$. It is known that $1$, $3$ and $9$ have the given property. Assume that $d$ is a $k$ digit number such that whenever $d$ divides an integer $n$, $d$ will also divide any integer $m$ having the same digits as $n$. Then there exists a $k+2$ digit number $10a_1a_2\dots a_k$ which is divisible by $d$. Hence $a_1a_2\dots a_k10$ and $a_1a_2\dots a_k01$ are also divisible by $d$. Since $a_1a_2\dots a_k10 - a_1a_2\dots a_k01 = 9$, $d$ divides $9$, and hence $d=1$, $d=3$ or $d=9$ as stated.