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A positive integer has $1001$ digits all of which are $1$'s. When this number is divided by $1001$ find the remainder.

I tried to think on it but couldn't get through. Please help.

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$$A=111111... (1001 \text { times})$$ $$A= 10^{1000}+10^{999}+10^{998}+\cdots +10^0$$ $$A= (10^{1000}+10^{997})+(10^{999}+10^{996})+\cdots+ (10^4+10^1)+ (10^3+10^0)+10^2$$ Now, any number of the form $10^{m+3}+10^m (m\geq 0)$ is divisible by $1001$. $$A=1001n+10^2$$ So the remainder is $100$.

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    $\begingroup$ This is incorrect, but you could get your approach to work by saying that $10^{1000}+10^{997}+10^{999}+10^{996}+10^{998}+10^{995}+10^{994}+10^{991}+...+10^4+10^1+10^3+10^0$ is divisible by $1001$, leaving remainder $10^2$, because $10^{m+\color{red}3}+10^m$ is divisible by $1001$ $\endgroup$ – J. W. Tanner Jul 6 at 3:52
  • $\begingroup$ @DwijRajHari Your answer is wrong. $\endgroup$ – Shreyansh Kuntal Jul 6 at 3:52
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    $\begingroup$ @J.W.Tanner I have corrected it. Thanks for pointing out the mistake. $\endgroup$ – user803121 Jul 6 at 4:04
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    $\begingroup$ To much is hidden in the „...“. In the Last Version of your answer you got the Result 1, now you get 100 and the answers Look like rather similar. You should find a notation That helps to avoid the error you did. $\endgroup$ – miracle173 Jul 6 at 4:36
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    $\begingroup$ @miracle173 the error was there because earlier I took $10^{m+2}+10^m$ to be divisible by $1001$ , whitch was wrong. It should be $10^{m+\color{red}3}+10^m$ $\endgroup$ – user803121 Jul 6 at 8:11
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$10^3\equiv-1\bmod1001$

$10^{999}\equiv-1\bmod1001$

$10^{1001}\equiv-100\equiv901\bmod1001$

$10^{1001}-1\equiv900\bmod1001$

$\dfrac{10^{1001}-1}9\equiv100\bmod1001$

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this question can actually done in a much simpler way guys! we know that, 1001 can perfectly divide 111111(a number with 6 one's) so in the given number,

                   *there are 1001 digits in which 996(the last no. divisible by 6)
                   can be divided with no remainder.*
                

therefore the last 5 one's (11111) must be divided by 1001 to get the remainder. 11111/1001 gives 100 as ur answer.

and u have the way to go!!

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