Number Theory

We can write a handy equation to solve this problem. 

First, let's note that every number satsifying the conditions given can be written in the form

$171x+5$ where x is an integer. 

We set this to equal 1000 to get the largest possible x can be, so we have

$171x+5 = 1000$

We are trying to solve for x right now. We have

$171x=995$

Dividing both sides, we find that 

$x=5.81871345029$

We must round down, so 5 must be our answer. 

So our final answer is 5. 

Thanks! :)