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How many numbers between $1000$ and $2000$ leave a remainder of $3$ when divided by $21?$

 Jun 20, 2024

Best Answer 

 #1
avatar+42 
+1

First, we look at 1000 mod 21 and 2000 mod 21. 1000 is congruent to 13 mod 21, and 2000 is congruent to 5 mod 21. We are trying to find how many numbers between 1000 and 2000 are congruent to 3 mod 21. The greatest number is 21*95+3=1998 and the least number is 21*48+3=1011. Because it cycles every 21 numbers, there are 95-48+1=\(48\) numbers that satisfy this property

 

Feel free to tell me if I did anything wrong! :D

 Jun 20, 2024
 #1
avatar+42 
+1
Best Answer

First, we look at 1000 mod 21 and 2000 mod 21. 1000 is congruent to 13 mod 21, and 2000 is congruent to 5 mod 21. We are trying to find how many numbers between 1000 and 2000 are congruent to 3 mod 21. The greatest number is 21*95+3=1998 and the least number is 21*48+3=1011. Because it cycles every 21 numbers, there are 95-48+1=\(48\) numbers that satisfy this property

 

Feel free to tell me if I did anything wrong! :D

Tottenham10 Jun 20, 2024

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