<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] Observe that $f(x^5) = f(x)(x^{16}-x^{15}+2x^{11}-2x^{10}+3x^6-3x^5+4x-4) + 5$.
So the remainder will be $5$.
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