Find the Remainder when $24242424$.... upto $300$ digits is divided by $999$?


I did grouping of 3 digit starting from right and added them and I stopped till my number gets less than $999$.


Why I am getting wrong Ans?Please correct me if I am wrong?

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    $\begingroup$ I can make no sense of what you did, I’m afraid. Can you explain your reasoning? You might want to look at the answers to this very similar question for ideas. $\endgroup$ – Brian M. Scott Nov 3 '15 at 4:44
  • $\begingroup$ Um, 24242424... 300 doesn't equal 2424242400 which doesn't equal 1068. $\endgroup$ – fleablood Nov 3 '15 at 4:44
  • $\begingroup$ Um... your explanation doesn't make any sense. $\endgroup$ – fleablood Nov 3 '15 at 4:52
  • $\begingroup$ @fleablood I followed the question this :math.stackexchange.com/questions/1508995/… and did wrong. $\endgroup$ – Jack Nov 3 '15 at 4:59
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    $\begingroup$ Where are these questions coming from? This same kind of question, in the same style, came up yesterday at math.stackexchange.com/questions/1508995/… $\endgroup$ – alex.jordan Nov 3 '15 at 6:03

Since $1000\equiv1$ mod $999$, then you have $$424\cdot1000^0+\cdot242\cdot1000^1+424\cdot1000^2+\cdots\equiv424+242+424+\cdots$$ So it's the same as $50$ copies of $424+242=666$.

Now reduce $50\cdot666=48\cdot666+2\cdot666\equiv2\cdot666\equiv333$.

  • $\begingroup$ @alex.jorden 50⋅666=48⋅666+2 your last step is not understood by me?Why you reduced $\endgroup$ – Jack Nov 3 '15 at 8:57
  • $\begingroup$ Your approach is right.But Why I am wrong?A similar question was explained here in your comments $\endgroup$ – Jack Nov 3 '15 at 9:00
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    $\begingroup$ @Jack The 48 has a factor of $3$ in it, and the $666$ has a factor of $333$. So together, that part is divisible by $999$, and therefore equivalent to $0$. $\endgroup$ – alex.jordan Nov 3 '15 at 17:51

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