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Try this beautiful problem from the PRMO, 2010 based on Divisibility.

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

Inequalities

Algebra

Number Theory

But try the problem first...

Answer: is 48.

Source

Suggested Reading

PRMO, 2019, Question 8

Elementary Number Theory by David Burton

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.

- https://www.cheenta.com/smallest-perimeter-of-triangle-aime-2015-question-11/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s

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