AIME-2001

[Arif Ahmed]

How many positive integer multiples of $1001$ can be expressed in the form $10^j-10^i$.where $i$ and $j$ are integers and $0 \leq i<j \leq 99$

[shehab ahmed]

note that $1001=7*11*13$
as $j>i$ we have $10^{j-i}$ is congruent to $1 \pmod 7,11,13$
find the order of $10 \pmod {7,11,13}$ these numbers and use the following lemma
if b is the order of $a \pmod c$ and $d$ is any positive integer such that $a^d \equiv 1 \pmod c$ then, $b \mid d$