+743 How many of the $1000$ smallest positive integers are congruent to $5$ modulo $171?$
+1876 We can list out all numbers since there are not that many.
5 modulo 171 means that the number leaves a remainder of 5 when divided by 171.
First, we have 5, since we have \(171*0 + 5 = 5\)
We add 171 to every number from here on.
So, we have
\(5+171=176\\ 176+171 = 347\\ 347+171 = 518\\ 518+171=689\\ 689+171=860\\\)
Adding anymore would result in a number greater than 1000.
Thus, we have \(6\) numbers.
So our answer is 6.
Thanks! :)
