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Problem 754

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Product of Gauss Factorials

The Gauss Factorial of a number $n$ is defined as the product of all positive numbers $\leq n$ that are relatively prime to $n$. For example $g(10)=1\times 3\times 7\times 9 = 189$.

Also we define
$$\displaystyle G(n) = \prod_{i=1}^{n}g(i)$$

You are given $G(10) = 23044331520000$.

Find $G(10^8)$. Give your answer modulo $1\ 000\ 000\ 007$.

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高斯阶乘之积

整数$n$的高斯阶乘是指所有小于等于$n$且与$n$互质的正整数的乘积。例如，$g(10)=1\times 3\times 7\times 9 = 189$。

记
$$\displaystyle G(n) = \prod_{i=1}^{n}g(i)$$

已知$G(10) = 23044331520000$。

求$G(10^8)$，并将你的答案对$1\ 000\ 000\ 007$取余。

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