Beautiful polynomial

Since $P(1)=P(3)=P(5)=P(7)=a$ , there exist a polynomial $Q(x)$ with integer coefficients and $P(x)=Q(x)(x-1)(x-3)(x-5)(x-7)+a$ Put $x=2, 4, 6, 8$ in above equality, we get: $-2a=-15Q(2)=9Q(4)=-15Q(6)=105Q(8)$ .

From this, $2a$ is divisible by $15,9$ , and $105$ , so $a$ is divisible by $315$ .

Hence, the minimum positive value of $a$ is $315$ .

The equality holds iff $Q(2)=Q(6)=42,Q(4)=-70,Q(8)=-6$ .

We can choose $Q(x)=-8x^3+124x^2-576x+762$ and $P(x)=(-8x^3+124x^2-576x+762)(x-1)(x-3)(x-5)(x-7)+315$ .