# A Groovy Number Theory Problem [closed]

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?

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

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• How have you tried to solve this problem? (Hint: consider modular arithmetic.) – Robert Shore Feb 14 at 20:05
• 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. – Murv___J Feb 14 at 20:17
• 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. – Robert Shore Feb 14 at 20:18
• And can you solve $n(2n+1)\equiv 2 \pmod 9$? – lulu Feb 14 at 20:30
• @Peter Yes. $\quad$ – lulu Feb 14 at 21:35