## [ Divisibility of numbers. General properties ] [ Factorization ]

Does the polynomial $1+x^{4}+x^{8}+\ldots+x^{4 k}$ divide the polynomial $1+x^{2}+x^{4}+\ldots+x^{2 k}$ ?

#
$\frac{1+x^{4}+\ldots+x^{4 k}}{1+x^{2}+\ldots+x^{2 k}}=\frac{x^{4 k+4}-1}{x^{4}-1}: \frac{x^{2 k+2}-1}{x^{2}-1}=\frac{x^{2 k+2}+1}{x^{2}+1}=\frac{\left(x^{2}\right)^{k+1}+1}{x^{2}+1}$. If $k+1$ is odd, the numerator is divisible by the denominator, and if

$k+1=2 n$, when dividing, the remainder is 2, since $x^{2}(k+1)-1=x^{4 n}-1$ is divisible by $x^{4}-1$, and thus by $x^{2}+1$

## Answer

When $k$ is even, it is divisible, and when $k$ is odd, it is not.