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Olympiad workbook
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IOQM
Chapter
Complex Number

Olympiad workbook · IOQM · Complex Number

How many positive integers n are there such that 3 ≤ n ≤ 100 and x^...

How many positive integers nn are there such that 3≤n≤1003 \leq n \leq 100 and x2n+x+1x^{2^n}+x+1 is divisible by x2+xx^2+x + 1 ?

Correct answer

49

Step-by-step solution

Since ω\omega and ω2\omega^2 are factors of X2+X+1\mathrm{X}^2+\mathrm{X}+1; so ω\omega and ω2\omega^2 will be factors of X2n+X+1⇒2n\mathrm{X}^{2 \mathrm{n}}+\mathrm{X}+1 \Rightarrow 2^n must be of the form 3k+2;k∈I3 k+2 ; k \in I. For this to happen, nn must be odd. So n={3,5,7,……99}\mathrm{n}=\{3,5,7, \ldots \ldots 99\} Number of such numbers =49=49
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