Number Theory

We can write an equation to solve this problem. 

First, let's note that every number in the form 5 modulo 171 can be written in the expression

$171x+5$ where x is an integer. 

We set this to equal 1000 to get the largest number possible. we have

$171x+5 = 1000$

We are trying to find x. We have

$171x=995$

Dividing, we get

$x=5.81871345029$

We actually have to round down, so we get x is 5. 

So our final answeer is 5. 

Thanks! :)