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How many of the $1000$ smallest positive integers are congruent to $5$ modulo $171?$

 Jul 11, 2024
 #1
avatar+1892 
+1

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! :)

 Jul 11, 2024

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