<s>[INST] Consider a function $f(x) = x^4+x^3+x^2+x+1$, where x is an integer, $x\gt 1$. What will be the remainder when $f(x^5)$ is divided by $f(x)$? Consider a function $f(x) = x^4+x^3+x^2+x+1$, where x is an integer, $x\gt 1$. What will be the remainder when $f(x^5)$ is divided by $f(x)$ ?
$f(x)=x^4+x^3+x^2+x+1$
$f(x^5)=x^{20}+x^{15}+x^{10}+x^5+1$
 [/INST] Rewrite the expression as 
$$(x^{20}-1) +(x^{15}-1)+(x^{10}-1)+(x^5-1)+1+4.$$
The expression $x^4+x^3+x^2+x+1$ divides the first four terms, since $x^5-1$ does. So the remainder is $5$.  
The idea obviously generalizes. 
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