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.$
Inequalities
Algebra
Number Theory
But try the problem first...
Answer: is 48.
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.
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