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Algebra Arithmetic Math Olympiad PRMO

Divisibility Problem | PRMO 2019 | Question 8

Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2015 based on Smallest Perimeter of Triangle.

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


Answer: is 48.

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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