# Prove that x^4+x^3+x^2+x+1∣x^(4n)+x^(3n)+x^(2n)+x^n+1

kadejoset 2022-07-18 Answered
Prove that ${x}^{4}+{x}^{3}+{x}^{2}+x+1\mid {x}^{4n}+{x}^{3n}+{x}^{2n}+{x}^{n}+1$
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## Answers (1)

Eve Good
Answered 2022-07-19 Author has 18 answers
${X}^{4}+{X}^{3}+{X}^{2}+X+1=\frac{{X}^{5}-1}{X-1}$
${X}^{4n}+{X}^{3n}+{X}^{2n}+{X}^{n}+1=\frac{{X}^{5n}-1}{{X}^{n}-1}$
Now, use the fact that ${X}^{5}-1|{X}^{5n}-1$ and that for n not divisible by 5 we have
$gcd\left({X}^{5}-1,{X}^{n}-1\right)={X}^{gcd\left(5,n\right)}-1=X-1$
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