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 $1\,000\,000\,007$.
Solution
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