Difference between revisions of "2017 AIME I Problems/Problem 9"
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Writing out the recursive statement for <math>a_n, a_{n-1}, \dots, a_{10}</math> and summing them gives <cmath>a_n+\dots+a_{10}=(a_n+\cdots+a_{11})+a_{10}=100(a_{n-1}+\dots+a_{10})+n+\dots+10</cmath> | Writing out the recursive statement for <math>a_n, a_{n-1}, \dots, a_{10}</math> and summing them gives <cmath>a_n+\dots+a_{10}=(a_n+\cdots+a_{11})+a_{10}=100(a_{n-1}+\dots+a_{10})+n+\dots+10</cmath> | ||
Which simplifies to <cmath>a_n=99(a_{n-1}+\dots+a_{10})+\frac{1}{2}(n+10)(n-9)</cmath> | Which simplifies to <cmath>a_n=99(a_{n-1}+\dots+a_{10})+\frac{1}{2}(n+10)(n-9)</cmath> | ||
− | Therefore, <math>a_n</math> is divisible by 99 if and only if <math>\frac{1}{2}(n+10)(n-9)</math> is divisible by 99, so <math>(n+10)(n-9)</math> needs to be divisible by 9 and 11. Assume that <math>n+10</math> is a multiple of 11. Writing out a few terms, <math>n=12, 23, 34, 45</math>, we see that <math>n=45</math> is the smallest <math>n</math> that works in this case. Next, assume that <math>n-9</math> is a multiple of 11. Writing out a few terms, <math>n=20, 31, 42, 53</math>, we see that <math>n=53</math> is the smallest <math>n</math> that works in this case. The smallest <math>n</math> is <math>\boxed{045}</math>. | + | Therefore, <math>a_n</math> is divisible by 99 if and only if <math>\frac{1}{2}(n+10)(n-9)</math> is divisible by 99, so <math>(n+10)(n-9)</math> needs to be divisible by 9 and 11. Since our goal is just finding the minimum <math>n</math>, we can do casework on whether the <math>n+10</math> or <math>n-9</math> is a multiple of 11 (We choose 11 because it is prime). Assume that <math>n+10</math> is a multiple of 11. Writing out a few terms, <math>n=12, 23, 34, 45</math>, we see that <math>n=45</math> is the smallest <math>n</math> that works in this case. Next, assume that <math>n-9</math> is a multiple of 11. Writing out a few terms, <math>n=20, 31, 42, 53</math>, we see that <math>n=53</math> is the smallest <math>n</math> that works in this case. The smallest <math>n</math> is <math>\boxed{045}</math>. |
− | Note that we can also construct the solution using CRT by assuming either <math>11</math> divides <math>n+10</math> and <math>9</math> divides <math>n-9</math>, or <math>9</math> divides <math>n+10</math> and <math>11</math> divides <math>n-9</math>, and taking the smaller solution. | + | Note that we can also construct the solution using CRT by assuming either <math>11</math> divides <math>n+10</math> and <math>9</math> divides <math>n-9</math>, or <math>9</math> divides <math>n+10</math> and <math>11</math> divides <math>n-9</math>, and taking the smaller solution. (We know this because it's impossible for both <math>n+10</math> and <math>n-9</math> to be divisible by 3). |
==Another way to get the quadratic== | ==Another way to get the quadratic== | ||
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-mathkiddus | -mathkiddus | ||
− | ==Solution 2 | + | ==Solution 2== |
<cmath>a_n \equiv a_{n-1} + n \pmod {99} </cmath> | <cmath>a_n \equiv a_{n-1} + n \pmod {99} </cmath> | ||
By looking at the first few terms, we can see that | By looking at the first few terms, we can see that | ||
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The smallest positive integer solution greater than <math>10</math> is <math>n=\boxed{045}</math>. | The smallest positive integer solution greater than <math>10</math> is <math>n=\boxed{045}</math>. | ||
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==Question== | ==Question== | ||
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<b>Answer:</b> Yes. | <b>Answer:</b> Yes. | ||
− | We will solve the question by looking for solutions for <math>m\in[- | + | We will solve the question by looking for solutions for <math>m\in[-49,-48,\cdots,-1,0,1,\cdots,48,49]</math>. Let the square be <math>m</math>, thus satisfying <math>m^2-64\equiv 0\pmod{99}</math> and <math>m^2-64=(m+8)(m-8)=99k</math> for some integer <math>k</math>. Checking the first factor, <math>(m+8)</math>, for factors of <math>11</math> yields: |
<math>m+8=0\Rightarrow m-8=-16</math> | <math>m+8=0\Rightarrow m-8=-16</math> | ||
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Now we check the second factor, <math>(m-8)</math>: | Now we check the second factor, <math>(m-8)</math>: | ||
− | <math>m-8=0\Rightarrow m | + | <math>m-8=0\Rightarrow m+8=16</math> |
− | <math>m-8=11\Rightarrow m | + | <math>m-8=11\Rightarrow m+8=27</math> |
− | <math>m-8=22\Rightarrow m | + | <math>m-8=22\Rightarrow m+8=38</math> |
− | <math>m-8=33\Rightarrow m | + | <math>m-8=33\Rightarrow m+8=49</math> |
− | <math>m-8=44\Rightarrow m | + | <math>m-8=44\Rightarrow m+8=60</math> |
We immediately see that <math>m=8</math> is a solution. Using a similar argument as above for the others, we notice that <math>m=19</math> is the only solution in this group. Using the property that <math>ab\equiv(-a)(-b)\mod99</math>, it is clear that <math>m=-19</math> is also a solution. As a result, <math>m=\pm8</math> and <math>m=\pm19</math> are the only solutions. (This process takes a much shorter time than it seems.) | We immediately see that <math>m=8</math> is a solution. Using a similar argument as above for the others, we notice that <math>m=19</math> is the only solution in this group. Using the property that <math>ab\equiv(-a)(-b)\mod99</math>, it is clear that <math>m=-19</math> is also a solution. As a result, <math>m=\pm8</math> and <math>m=\pm19</math> are the only solutions. (This process takes a much shorter time than it seems.) |
Latest revision as of 21:24, 8 December 2024
Contents
Problem 9
Let , and for each positive integer let . Find the least positive such that is a multiple of .
Solution 1
Writing out the recursive statement for and summing them gives Which simplifies to Therefore, is divisible by 99 if and only if is divisible by 99, so needs to be divisible by 9 and 11. Since our goal is just finding the minimum , we can do casework on whether the or is a multiple of 11 (We choose 11 because it is prime). Assume that is a multiple of 11. Writing out a few terms, , we see that is the smallest that works in this case. Next, assume that is a multiple of 11. Writing out a few terms, , we see that is the smallest that works in this case. The smallest is .
Note that we can also construct the solution using CRT by assuming either divides and divides , or divides and divides , and taking the smaller solution. (We know this because it's impossible for both and to be divisible by 3).
Another way to get the quadratic
Writing out the first couple of terms modulo , we find so we have We can compute their finite differences, Since there are 3 rows we know it is a quadratic and we can continue, by finding the quadratic passing through to get -mathkiddus
Solution 2
By looking at the first few terms, we can see that This implies Since , we can rewrite the equivalence, and simplify The only squares that are congruent to are and , so yields as the smallest integer solution.
yields as the smallest integer solution.
yields as the smallest integer solution.
yields as the smallest integer solution. However, must be greater than .
The smallest positive integer solution greater than is .
Question
Is there an efficient way to notice that the only squares that are congruent to are and ?
Answer: Yes.
We will solve the question by looking for solutions for . Let the square be , thus satisfying and for some integer . Checking the first factor, , for factors of yields:
In the first case, the product does divide , so is a solution. For the others, since the first factor already divides , the second factor must divide (or in the case of , which already has a factor of ) in order for the product to divide . Here, that is not the case, so is the only possible solution.
Now we check the second factor, :
We immediately see that is a solution. Using a similar argument as above for the others, we notice that is the only solution in this group. Using the property that , it is clear that is also a solution. As a result, and are the only solutions. (This process takes a much shorter time than it seems.)
~eevee9406
Solution 3
. Using the steps of the previous solution we get up to . This gives away the fact that so either or must be a multiple of 9.
Case 1 (): Say and after simplification .
Case 2: (): Say and after simplification .
As a result must be a divisor of and after doing some testing in both cases the smallest value that works is .
~First
Solution 4 (not good, risky)
We just notice that , so we are just trying to find modulo , or modulo . Also, the sum to is divisible by , and is the first one that is. Thus, if we sum to the is cut off and thus is just a sum to .
Without checking whether there are other sums congruent to , we can just write the answer to be .
Solution 5
Let . We can find a formula for :
.
Notice that both can't have a factor of 3. Thus we can limit our search range of n to . Testing values for n in our search range (like 7,8,16,17,25,26...), we get that 35 is the least n. But, don't write that down! Remember, , so, the 35th term in b corresponds to the 45th term in a. Thus our answer is .
-AlexLikeMath
Solution 6 (slower and safer bash)
The first thing you should realize is that each term after the tenth is another two-digit number chained to the last number. . Now the fact that the sequence starts at can be completely discarded for this solution. Just consider then same as , and we can add nine to the answer at the end.
The second step is to split as and and solve for divisibility rules individually. Let's start with because it gives us the most information to continue.
In any number generated, if the numbers don't go beyond , then the highest number we can get is , with every odd digit being . This is a little risky because we are assuming that it doesn't exceed . If someone wanted to be absolutely sure they could continue, but this is unnecessary later and a big hassle. Anyways, now we write an equation to check for divisibility by . The expression being .
The concept here is to add to the term altogether, then subtract the number of ones in it, which is . Simplify to congruent to . Now notice the divide by two can be discarded because one of or will be even. So if or is to be divisible by , we can make a simple list.
Now we test each for divisibility by . This is done by making a list that ultimately calculates the sum of every digit in the large number. to has the first digit . to has the first digit , and so on. The necessary thing to realize is that the sum of all digits is divisible by , so we only have to solve for the sum of the first digits, and then the short list of second digits.
For example, let's test .
So we know that include both and , so that's right away. contains numbers that have the first digit , so . Then we add together, which is . , which is not divisible by , so it is not the answer.
Do this for just a minute you get that sums to , a multiple of nine! So is the answer, right? Don't forget we have to add because we translated to at the very beginning! Finally, after a short bash, we get .
-jackshi2006 ( by PureSwag)
Solution 7 (Timekilling plain Bash!!!!!!!!!!!!!!!!!)
We will work. The recursive formula now becomes . Now, we will bash. For convenience, everything is taken . The sequence is Hence the least number is .
~typos fixed by lpieleanu
Solution 8
Taking the recurrence mod , we have for . Then, we have Then, we simply can test these values of to find that produces a value that is also divisible by .
-A1001
Note
By the way, if you're wondering, is the -digit number The prime factorization of is and is the -digit number
See also
2017 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.