Difference between revisions of "2021 Fall AMC 12B Problems/Problem 25"
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==Solution 4== | ==Solution 4== | ||
− | Upon adding one to <math>n</math>, consider each individual remainder. Either it will increase by 1, or it will "wrap around | + | Upon adding one to <math>n</math>, consider each individual remainder. Either it will increase by 1, or it will "wrap around', going from <math>5\rightarrow 0</math> mod 6 and in general, <math>n-1 \rightarrow 0</math> mod n. We will use '<math>+1</math>' to refer to remainders that increase by 1, and 'wrap-arounds' to refer to remainders that go to 0. |
− | Clearly, < | + | Clearly, <math>9</math> <math>+1</math>s isn't possible, since then <math>R(n)\ne R(n+1)</math>. |
− | If there are < | + | If there are <math>8</math> <math>+1</math>s and <math>1</math> wrap-arounds, the wrap-around must be equal to <math>-8</math>, which is the case for <math>(9)</math>. However, if <math>n</math> is <math>8</math> mod <math>9</math>, it clearly must also be <math>2</math> mod <math>3</math>, meaning there must more be one wrap-around, and this case won't work. |
− | If there are < | + | If there are <math>7</math> <math>+1</math>s and <math>2</math> wrap-arounds, these two wrap-arounds must add to <math>-7</math>. For the possible modulo, we could have <math>(2,7)</math>, <math>(3,6)</math>, and <math>(4,5)</math>. Clearly, <math>(3,6)</math> won't work since if it is <math>5</math> mod <math>6</math>, then it must also be <math>1</math> mod <math>2</math>, meaning <math>(3,6)</math> won't be the only wrap-arounds. Similarly, <math>(4,5)</math> doesn't work since <math>3</math> mod <math>4</math> implies that <math>1</math> mod <math>2</math> will also be a wrap-around. That leaves <math>(2,7)</math>. The number must be <math>1</math> mod <math>2</math> and <math>6</math> mod <math>7</math>, or in other words, <math>-1</math> mod <math>2</math> and <math>-1</math> mod <math>7</math>, meaning n will be <math>-1 \equiv 13</math> mod <math>14</math>. Testing all such two digits numbers that are equivalent to <math>13</math> mod <math>14</math>, we see that <math>13</math> and <math>97</math> are the only two that work. |
− | If there are < | + | If there are <math>6</math> <math>+1</math>s and <math>3</math> wrap-arounds, the only possible combination of modulo is <math>(2,3,4)</math>. Thus, <math>n</math> must be <math>11</math> mod <math>12</math>. However, this means that mod <math>6</math> will also be a wrap around, so this case won't work. |
− | Notice that there can be no more cases, as for < | + | Notice that there can be no more cases, as for <math>5</math> <math>+1</math>s, no matter what mods wrap around, the <math>+1</math>s will not be able to balance them out, as it's magnitude is too small. Therefore, there are only <math>\boxed{\textbf{C) }2}</math> numbers. |
==Video Solution== | ==Video Solution== |
Revision as of 19:30, 4 November 2024
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)
Solution 3
To get from to , would add by for each remainder . However, given that some of these remainders can "round down" to given the nature of mods, we must calculate the possible values of such that the remainders in "rounds down" by a total of , effectively canceling out the adding by initially.
To do so, we will analyze the "rounding down" for each of :
: subtract by
: subtract by
: subtract by , but this also implies mod , so subtract by .
: subtract by
: subtract by , but this also implies mod and , so subtract by : too much
: subtract by
: subtract by , but this also implies mod and , so subtract by : too much
: subtract by , but this also implies mod , so subtract by : too much
: subtract by : too much
Notice that . By testing these sums, we can easily show that the only time when the total subtraction is is when AND . By CRT, :
As in solution 1, then, only and give possible solutions, so our answer is .
~xHypotenuse
Solution 4
Upon adding one to , consider each individual remainder. Either it will increase by 1, or it will "wrap around', going from mod 6 and in general, mod n. We will use '' to refer to remainders that increase by 1, and 'wrap-arounds' to refer to remainders that go to 0.
Clearly, s isn't possible, since then .
If there are s and wrap-arounds, the wrap-around must be equal to , which is the case for . However, if is mod , it clearly must also be mod , meaning there must more be one wrap-around, and this case won't work.
If there are s and wrap-arounds, these two wrap-arounds must add to . For the possible modulo, we could have , , and . Clearly, won't work since if it is mod , then it must also be mod , meaning won't be the only wrap-arounds. Similarly, doesn't work since mod implies that mod will also be a wrap-around. That leaves . The number must be mod and mod , or in other words, mod and mod , meaning n will be mod . Testing all such two digits numbers that are equivalent to mod , we see that and are the only two that work.
If there are s and wrap-arounds, the only possible combination of modulo is . Thus, must be mod . However, this means that mod will also be a wrap around, so this case won't work.
Notice that there can be no more cases, as for s, no matter what mods wrap around, the s will not be able to balance them out, as it's magnitude is too small. Therefore, there are only numbers.
Video Solution
~MathProblemSolvingSkills.com
Video Solution by Mathematical Dexterity
https://www.youtube.com/watch?v=Fy8wU4VAzkQ
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