Difference between revisions of "2024 AIME I Problems/Problem 13"
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==Solution 4== | ==Solution 4== | ||
− | These | + | These kinds of problems are, by nature, elementary. We get: <cmath> m^4 \equiv -1 \pmod{p^2},</cmath> thus, <math>m</math> is even, and <cmath>p^2 = 16k + 1</cmath> or <cmath>p = 16k + 1,</cmath> since <math>p</math> is prime. Therefore, the smallest possible such <math>p</math> is <math>17</math>. Again, |
− | or <cmath>p=16k+1</cmath> | + | <cmath>m^4 \equiv 16 \pmod{17}.</cmath> This is where it gets a bit tricky. |
− | <cmath>m^4\equiv | + | <cmath>(m^2 - 4)(m^2 + 4) \equiv 0 \pmod{17}</cmath> or <cmath>m^2 \equiv 13 \pmod{17}.</cmath> This gives rise to: <cmath>m \equiv 8 \pmod{17}.</cmath> |
− | <cmath>(m^2-4) | + | Now, <math>m</math> lies in the series <math>42, 76, 110, 144, \ldots</math>. It is easy to see that the smallest value of <math>m</math> is <math>110</math> as neither <math>42</math> nor <math>76</math> satisfy all criteria. |
− | Now m lies in the series 42, 76, 110, 144 | ||
~Grammaticus | ~Grammaticus |
Revision as of 17:21, 18 June 2024
Contents
[hide]Problem
Let be the least prime number for which there exists a positive integer such that is divisible by . Find the least positive integer such that is divisible by .
Solution 1
If
For integer
If
If
If
In conclusion, the smallest possible
Solution by Quantum-Phantom
Solution 2
We work in the ring
Solution 3 (Easy, given specialized knowledge)
Note that means The smallest prime that does this is and for example. Now let be a primitive root of The satisfying are of the form, So if we find one such , then all are Consider the from before. Note by LTE. Hence the possible are, Some modular arithmetic yields that is the least value.
~Aaryabhatta1
Solution 4
These kinds of problems are, by nature, elementary. We get: thus, is even, and or since is prime. Therefore, the smallest possible such is . Again, This is where it gets a bit tricky. or This gives rise to: Now, lies in the series . It is easy to see that the smallest value of is as neither nor satisfy all criteria.
~Grammaticus
Video Solution
https://www.youtube.com/watch?v=_ambewDODiA
~MathProblemSolvingSkills.com
Video Solution 1 by OmegaLearn.org
Video Solution 2
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See also
2024 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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.