2021 Fall AMC 12B Problems/Problem 25
Contents
Problem
For a positive integer, let be the sum of the remainders when is divided by , , , , , , , , and . For example, . How many two-digit positive integers satisfy
Solution 1
Note that we can add to to get , but must subtract for all . Hence, we see that there are four ways to do that because . Note that only is a plausible option, since indicates is divisible by , indicates that is divisible by , indicates is divisible by , and itself indicates divisibility by , too. So, and is not divisible by any positive integers from to , inclusive, except and . We check and get that only and give possible solutions so our answer is .
- kevinmathz
Solution 2
Denote by the remainder of divided by . Define .
Hence,
Hence, this problem asks us to find all , such that .
: .
We have .
Therefore, there is no in this case.
: and .
The condition implies . This further implies . Hence, .
To get , we have .
However, we have .
Therefore, there is no in this case.
: for and .
The condition implies with . Hence, and .
To get , we have .
However, we have .
Therefore, there is no in this case.
: for and .
To get , we have .
Hence, we must have and for .
Therefore, .
: for and .
The condition implies with . Hence, and .
To get , we have .
However, we have .
Therefore, there is no in this case.
: for and .
To get , we have .
This can be achieved if , , .
However, implies . This implies . Hence, . We get a contradiction.
Therefore, there is no in this case.
: for and .
The condition implies with . Hence, .
To get , we have . This implies .
Because and , we have . Hence, . However, in this case, we assume . We get a contradiction.
Therefore, there is no in this case.
: for and .
To get , we have . This is infeasible.
Therefore, there is no in this case.
: for .
To get , we have . This is infeasible.
Therefore, there is no in this case.
Putting all cases together, the answer is .
~Steven Chen (www.professorchenedu.com)
Video Solution
~MathProblemSolvingSkills.com
Video Solution by Mathematical Dexterity
https://www.youtube.com/watch?v=Fy8wU4VAzkQ
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