<s>[INST] High degree polynom divisibility The problem is:
Show that $x^{44}+x^{33}+x^{22}+x^{11}+1$ is divisible by $x^4+x^3+x^2+x+1$
I am not sure how to approach this problem so any help would be very appreciated. Thank you in advance.
 [/INST] HINT:
Both are in Geometric Series
$$1+x+x^2+x^3+x^4=\frac{1-x^5}{1-x}$$ and $$1+x^{11}+x^{22}+x^{33}+x^{44}=\frac{1-x^{55}}{1-x^{11}}$$
So, $$\frac{1+x^{11}+x^{22}+x^{33}+x^{44}}{1+x+x^2+x^3+x^4}=\frac{(1-x^{55})(1-x)}{(1-x^{11})(1-x^5)}$$
Now using Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$, 
$1-x^{55}$ is divisible by $1-x^5,1-x^{11}$ and 
$(1-x^{11},1-x^5)=1-x$
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