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Problem 778

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Freshman’s Product

If $a,b$ are two nonnegative integers with decimal representations $a=(\dots a_2a_1a_0)$ and $b=(\dots b_2b_1b_0)$ respectively, then the freshman’s product of $a$ and $b$, denoted $a\boxtimes b$, is the integer $c$ with decimal representation $c=(\dots c_2c_1c_0)$ such that $c_i$ is the last digit of $a_i\cdot b_i$.

For example, $234 \boxtimes 765 = 480$.

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

For example, $F(2, 7) = 204$, and $F(23, 76) \equiv 5870548 \pmod{1\ 000\ 000\ 009}$.

Find $F(234567,765432)$, give your answer modulo $1\ 000\ 000\ 009$.

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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$a=(\dots a_2a_1a_0)$和$b=(\dots b_2b_1b_0)$，则$a$和$b$的初学者乘积，记为$a\boxtimes b$，是另一个整数$c$，其十进制表示为$c=(\dots c_2c_1c_0)$，其中$c_i$是$a_i\cdot b_i$的末位数字。

例如，$234 \boxtimes 765 = 480$。

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

例如，$F(2, 7) = 204$，$F(23, 76) \equiv 5870548 \pmod{1\ 000\ 000\ 009}$。

求$F(234567,765432)$，并将你的答案对$1\ 000\ 000\ 009$取余。

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