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Try this beautiful problem from the PRMO, 2010 based on Divisibility.
Divisibility Problem – PRMO 2019
Find the number of positive integers such that \(3 \leq n \leq 98\) and \(x^{2^{n}}+x+1\) is divisible by $ x^{2}+x+1.$
- is 107
- is 48
- is 840
- cannot be determined from the given information
Key Concepts
Inequalities
Algebra
Number Theory
Check the Answer
But try the problem first…
Answer: is 48.
Source
Suggested Reading
PRMO, 2019, Question 8
Elementary Number Theory by David Burton
Try with Hints
First hint
for n={3,5,…97}
Second Hint
where n is odd since factor of \(x^{2}+x+1\) is also factor of given expression
Final Step
Then n=48.
Other useful links
- https://www.cheenta.com/smallest-perimeter-of-triangle-aime-2015-question-11/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s
Related Program
Math Olympiad Program… taught by Olympians and researchers