Difference between revisions of "2024 AIME I Problems/Problem 13"
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Let <math>p</math> be the least prime number for which there exists a positive integer <math>n</math> such that <math>n^{4}+1</math> is divisible by <math>p^{2}</math>. Find the least positive integer <math>m</math> such that <math>m^{4}+1</math> is divisible by <math>p^{2}</math>. | Let <math>p</math> be the least prime number for which there exists a positive integer <math>n</math> such that <math>n^{4}+1</math> is divisible by <math>p^{2}</math>. Find the least positive integer <math>m</math> such that <math>m^{4}+1</math> is divisible by <math>p^{2}</math>. | ||
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If | If |
Revision as of 13:13, 5 February 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 3
We work in the ring
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.