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Problem 778

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Freshman’s Product

If $a,b$ are two nonnegative integers with decimal representations $a=(\dots a_2{a}_1{a}_0)$ and $b=(\dots b_2{b}_1{b}_0)$ respectively, then the freshman’s product of $a$ and $b$, denoted $a⊠b$, is the integer $c$ with decimal representation $c=(\dots c_2{c}_1{c}_0)$ such that $c_i$ is the last digit of $a_i\cdot b_i$.

For example, $234⊠765=480$.

Let $F(R,M)$ be the sum of $x_1⊠\cdots ⊠x_R$ for all sequences of integers $(x_1,\dots ,x_R)$ with $0\le x_i\le M$.

For example, $F(2,7)=204$, and $F(23,76)\equiv 5870548(\mathrm{mod}1000000009)$.

Find $F(234567,765432)$, give your answer modulo $1000000009$.

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初学者乘积

若$a$和$b$是两个非负整数，其十进制表示分别是$a=(\dots a_2{a}_1{a}_0)$和$b=(\dots b_2{b}_1{b}_0)$，则$a$和$b$的初学者乘积，记为$a⊠b$，是另一个整数$c$，其十进制表示为$c=(\dots c_2{c}_1{c}_0)$，其中$c_i$是$a_i\cdot b_i$的末位数字。

例如，$234⊠765=480$。

对于所有满足$0\le x_i\le M$的整数列$(x_1,\dots ,x_R)$，记$F(R,M)$为$x_1⊠\cdots ⊠x_R$之和。

例如，$F(2,7)=204$，$F(23,76)\equiv 5870548(\mathrm{mod}1000000009)$。

求$F(234567,765432)$，并将你的答案对$1000000009$取余。

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