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

 Jul 9, 2024
 #1
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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! :)

 Jul 9, 2024

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