+280 How many of the $1000$ smallest positive integers are congruent to $5$ modulo $171?$
+1892 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! :)
