+820 How many of the $1000$ smallest positive integers are congruent to $5$ modulo $171?$
+1804 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! :)
