Numbers whose Cyclic Permutations of 3-Digit Multiples are Multiples

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Theorem

Let $n$ be a two-digit positive integer with the following property:

Let $m$ be a $3$-digit multiple of $n$.
Then any permutation of the digits of $m$ is also a multiple of $n$.


Then $n$ is either $27$ or $37$.


Proof


Sources