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Turkey NMO 2008 1st Round - P06 (Number Theory)

Source:

August 24, 2012
modular arithmetic

Problem Statement

A positive integer nn is called a good number if every integer multiple of nn is divisible by nn however its digits are rearranged. How many good numbers are there?
<spanclass=latexbold>(A)</span> 3<spanclass=latexbold>(B)</span> 4<spanclass=latexbold>(C)</span> 6<spanclass=latexbold>(D)</span> 12<spanclass=latexbold>(E)</span> Infinitely many <span class='latex-bold'>(A)</span>\ 3 \qquad<span class='latex-bold'>(B)</span>\ 4 \qquad<span class='latex-bold'>(C)</span>\ 6 \qquad<span class='latex-bold'>(D)</span>\ 12 \qquad<span class='latex-bold'>(E)</span>\ \text{Infinitely many}