2024 AIME I Problems/Problem 13
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
Since
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
Where is the justification for why is even? If there is none, then this is just a "lucky solve"
~inaccessibles
Unless is even, which it is not,
has to be even. It is an elementary conclusion.
~
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 | ||
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