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

 Jun 27, 2024
 #1
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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! :)

 Jun 27, 2024

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