Question 21
Let $a\gt0,$ and let $P(x)$ be a polynomial with integer coefficients such that $$
\begin{aligned} &P(1)=P(3)=P(5) \\ &\quad {}=P(7)=a, \end{aligned}
$$ and $$
\begin{aligned} &P(2)=P(4)=P(6) \\ &\quad {}=P(8)=-a. \end{aligned}
$$ What is the smallest possible value of $a?$
\begin{aligned} &P(1)=P(3)=P(5) \\ &\quad {}=P(7)=a, \end{aligned}
$$ and $$
\begin{aligned} &P(2)=P(4)=P(6) \\ &\quad {}=P(8)=-a. \end{aligned}
$$ What is the smallest possible value of $a?$