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.

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