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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

But try the problem first…

Source

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.