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# Problem 635

## Subset sums

Let $A_q(n)$ be the number of subsets, $B$, of the set ${1,2,\ldots,q\cdot n}$ that satisfy two conditions:

1) $B$ has exactly $n$ elements;
2) the sum of the elements of $B$ is divisible by $n$.

E.g. $A_2(5)=52$ and $A_3(5)=603$.

Let $S_q(L)$ be $\sum A_q(p)$ where the sum is taken over all primes $p\le L$.
E.g. $S_2(10)=554$, $S_2(100) \mod 1\ 000\ 000\ 009 = 100433628$ and $S_3(100) \mod 1\ 000\ 000\ 009=855618282$.

Find $S_2(10^8)+S_3(10^8)$. Give your answer modulo $1\ 000\ 000\ 009$.

## 子集和

1) $B$中恰好有$n$个元素；
2) $B$中元素的和能够被$n$整除；