Question

a^5+a^4+a^3+a^2+a+1
_________________
a^6-1

Answer 1

$\displaystyle \frac{a^5+a^4+a^3+a^2+a+1}{a^6-1}= \frac{a^5+a^4+a^3+a^2+a+1}{(a^3)^2-1^2}=\\\\= \frac{a^5+a^4+a^3+a^2+a+1}{(a^3-1)(a^3+1)}= \frac{a^5+a^4+a^3+a^2+a+1}{(a-1)(a^2+a+1)(a+1)(a^2-a+1)}\\\\= \frac{a^5+a^4+a^3+a^2+a+1}{(a-1)(a^3+2a^2+2a+1)(a^2-a+1)}=\\\\= \frac{a^5+a^4+a^3+a^2+a+1}{(a-1)(a^5+a^4+a^3+a^2+a+1)}= \frac{1}{a-1}$