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

Question

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.

Answer 1

$$A=111111... (1001 \text { times})$$ $$A= 10^{1000}+10^{999}+10^{998}+\cdots +10^0$$ $$A= (10^{1000}+10^{997})+(10^{999}+10^{996})+\cdots+ (10^4+10^1)+ (10^3+10^0)+10^2$$ 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 $100$.

Answer 2

$10^3\equiv-1\bmod1001$

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

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

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

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

Answer 3

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!!