104. Divisibility condition. For which integer $a$ does the polynomial $x^{13}+x+90$ divide by $x^{2}-x+a ?$
Let $f(x)=x^{2}-x+a, g(x)=x^{13}+x+90$. Then $f(0)=$ $a, f(1)=a, g(0)=90, g(1)=92$. Therefore, the greatest common divisor of 90 and 92, which is 2, must divide $a$. Further, $f(-1)=$ $a+2, g(-1)=88$; hence $a$ is neither 1 nor $-2 ; f(-2)=a+6$, $g(-2)=-8104$, so $a \neq-1$. Therefore,

$$
\begin{gathered}
a=2^{\star} \quad \text { and } \frac{x^{13}+x+90}{x^{2}-x+2}= \\
=x^{11}+x^{10}-x^{9}-3 x^{8}-x^{7}+5 x^{6}+7 x^{5}- \\
-3 x^{4}-17 x^{3}-11 x^{2}+23 x+45
\end{gathered}
$$

[L. E. Bash, A. M. M., 71, 640 (June 1964).]