When 390390 is divided by either 77 or 1111, it gives the same non-zero remainder. How many integers from 11 to 20002000 that have this property?

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

When 390390 is divided by either 77 or 1111, it gives the same non-zero remainder. How many integers from 11 to 20002000 have this property?

Explore how to solve a modular arithmetic problem where 390390 gives the same remainder when divided by both 77 and 1111. This solution involves number theory concepts like the Chinese Remainder Theorem and is suitable for Grades 11-12 students.

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