We are tasked with finding how many integers between 11 and 20002000 satisfy the property that when divided by either 77 or 1111, they give the same non-zero remainder.

Step 1: Set Up the Remainder Equation

Let $N$ be an integer, and let $r$ be the common remainder when dividing $N$ by 77 and 1111. This means we can write:

$N \equiv r \ (\text{mod} \ 77) \quad \text{and} \quad N \equiv r \ (\text{mod} \ 1111)$

where