Number theory

Modulo is essentially the remainder we get when we divide two numbers.

For example, we have

$10 \equiv 1 (\mod 3)$

In this case, 1 is the remainder, 3 is the number we divide into 10. 

Thus, we say that 10 is congruent to 1 modulo 3. 

To solve the problem, we have the equation

$9n+1 \equiv 1 (\mod 9)$

We simply can write the equation

$9n+1 \leq 1000$

Now, we simplfy solve for n. Setting the two equations togther, we have

$9n = 999\\ n=111$

Thus, our answer is 111. 

Thanks! :)