30. Unknown remainder. Find the remainder of the division of $f\left(x^{5}\right)$ by $f(x)$, if

$$
f(x)=x^{4}+x^{3}+x^{2}+x+1
$$
30. Since $f(x)=x^{4}+x^{3}+x^{2}+x+1,(x-1) f(x)=x^{5}-1$. Further, $f\left(x^{5}\right)=\left(x^{20}-1\right)+\left(x^{15}-1\right)+\left(x^{10}-1\right)+\left(x^{5}-1\right)+4+1$. But $x^{5}-1$, and therefore $f(x)$, are divisors of each of the brackets. Hence,

$$
f\left(x^{5}\right)=[\text { multiple of } f(x)]+5
$$

[N. Anning, S. S. M., 54, 576 (October 1954).]