3
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Show that for every positive integers $k\geq n$, with $n$ not divisible by $3$, there is a positive integer divisible by $n$ and such that the sum of his digits is $k$.

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  • $\begingroup$ Do you have to find one pair of $n,k$ as in the title, or you have to prove "there exists ..." for all pairs of such $n, k$? $\endgroup$ – peterwhy Nov 23 '14 at 14:11
  • $\begingroup$ For a fixed n and every k there exists a positive integer $\endgroup$ – HeatTheIce Nov 23 '14 at 14:15
  • $\begingroup$ I asked this a couple of years agosto, ley me ser if O can find It. $\endgroup$ – Jorge Fernández Hidalgo Jan 28 '17 at 21:10
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IMO ShortList 1999, number theory problem 5 http://artofproblemsolving.com/community/c6h19765p131819

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