Q:

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

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.

A:

$10^3\equiv-1\bmod1001$
$10^{999}\equiv-1\bmod1001$
$10^{1001}\equiv-100\equiv901\bmod1001$
$10^{1001}-1\equiv900\bmod1001$
$\dfrac{10^{1001}-1}9\equiv100\bmod1001$

A:

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

