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

 Jul 9, 2024
 #1
avatar+1944 
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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

 

We are trying to find x. We have

 

Dividing, we get

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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