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

LiIIiam0216 Jun 20, 2024

#1**+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

Tottenham10 Jun 20, 2024

#1**+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