27. The product $1 \times 2 \times 3 \times \cdots \times n$ is denoted by $n!$. For example $4!=1 \times 2 \times 3 \times 4=24$. Let $M=1!\times 2!\times 3!\times 4!\times 5!\times 6!\times 7!\times 8!\times 9!$. How many factors of $M$ are perfect squares?
27. Ans: 672 .
$$
\begin{aligned}
M & =1!\times 2!\times 3!\times 4!\times 5!\times 6!\times 7!\times 8!\times 9! \\
& =2^{8} \times 3^{7} \times 4^{6} \times 5^{5} \times 6^{4} \times 7^{3} \times 8^{2} \times 9 \\
& =2^{30} \times 3^{13} \times 5^{5} \times 7^{3}
\end{aligned}
$$

A perfect square factor of $M$ must be of the form $2^{2 x} \times 3^{2 y} \times 5^{2 z} \times 7^{2 w}$, where $x, y, z$ and $w$ are whole numbers such that $2 x \leq 30,2 y \leq 13,2 z \leq 5,2 w \leq 3$. Hence, the number of perfect square factors of $M$ is $16 \times 7 \times 3 \times 2=672$.