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A number n is groovy iff the sum of the digits of n(2n+1) is 2018. Do such numbers exist and, if so, what are they?

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closed as off-topic by GNUSupporter 8964民主女神 地下教會, YiFan, Jyrki Lahtonen, José Carlos Santos, A. Pongrácz Feb 15 at 9:51

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    $\begingroup$ How have you tried to solve this problem? (Hint: consider modular arithmetic.) $\endgroup$ – Robert Shore Feb 14 at 20:05
  • $\begingroup$ Hello Robert. Thanks for the quick response. I've used modular arithmetic and have concluded that n(2n+1) is congruent to 2 mod 9. However, I'm uncertain of how to proceed from here. $\endgroup$ – Murv___J Feb 14 at 20:17
  • $\begingroup$ Try a different modulus and determine whether it's possible for a number of the form $n(2n+1)$ to have the necessary value in that modulus. $\endgroup$ – Robert Shore Feb 14 at 20:18
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    $\begingroup$ And can you solve $n(2n+1)\equiv 2 \pmod 9$? $\endgroup$ – lulu Feb 14 at 20:30
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    $\begingroup$ @Peter Yes. $\quad $ $\endgroup$ – lulu Feb 14 at 21:35