Problem 754

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

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

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

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

已知 $G (10) = 23044331520000$ 。

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