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How many numbers between $1000$ and $2000$ leave a remainder of $3$ when divided by $65?$

 Jun 23, 2024
 #1
avatar+85 
+1

just find how many numbers are divisible by 65 then add three and check if all of them work. 1040 is the least multiple of 65 that works and it is 16*65. 1950 is the greatest multiple of 65, and it is 30 *65. so there are 15 multiple of 65 between 1000 and 2000. we still need to check if adding three changes anything.  1953 is still in the range so the final answer would be \(\boxed{15}\). at least i think so  :P  ¯\_(ツ)_/¯

 Jun 23, 2024
 #2
avatar+1230 
+1

We can use an arithmetic sequence to solve this problem. 

The first number in this list is \(1043\), scine 1040 is divisble by 65. 

We just add 65 from there, until we get to 1953, since 1950 is divisble by 65. 

 

Thus, we have the sequence

\(1043, 1108, 1173...1953\)

 

Subtracting every term 3 and dividing by 65. we get

\(16,17,18,...30\)

 

Next, we subtract 15 from every term. 

We get

\(1,2,3...15\)

 

So our answer is 15...just like what Shmewy said!

 

Thanks! :)

 Jun 23, 2024
edited by NotThatSmart  Jun 23, 2024

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