Find the remainder when $x^{100}$ is divided by $x^8 - x^6 + x^4 - x^2 + 1.$ I need help in the problem:
Find the remainder when $x^{100}$ is divided by $x^8 - x^6 + x^4 - x^2 + 1.$
I have tried factoring $x^{100}-1$ and adding 1 to that, but that hasn't helped. Could someone please help with this?

$x^{100}-1=(x^{10}+1)(x^{90}-x^{80}+x^{70}-x^{60}+x^{50}-x^{40}+x^{30}-x^{20}+x^{10}-1)$
$=(x^8-x^6+x^4-x^2+1)(x^2+1)(x^{90}-x^{80}+x^{70}-x^{60}+x^{50}-x^{40}+x^{30}-x^{20}+x^{10}-1)$,
so $x^{100}=$
$(x^8-x^6+x^4-x^2+1)(x^2+1)(x^{90}-x^{80}+x^{70}-x^{60}+x^{50}-x^{40}+x^{30}-x^{20}+x^{10}-1) + 1.$
