8. Given $A \subseteq\{1,2, \cdots, 25\}$, satisfying: $\forall a, b \in A \quad a \neq b$, then $a b$ is not a perfect square, find the maximum value of $|A|$, and the number of $A$ that makes $|A|$ reach its maximum.

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8. Divide $\{1,2, \cdots, 25\}$ into the union of mutually disjoint subsets, such that the product of any two elements in the same subset is a perfect square, and the product of any two elements from different subsets is not a perfect square. Therefore, 
$$
\begin{array}{l}
\{1, 2, \cdots, 25\}=\bigcup_{i=1}^{16} A_{i}, A_{1}=\{1,4,9,16,25\}, A_{2}=\{2,8,18\}, A_{3}=\{3,12\}, \\
A_{4}=\{5,20\}, A_{5}=\{6,24\}, A_{6}=\{7\}, A_{7}=\{10\}, A_{8}=\{11\}, A_{9}=\{13\}, \\
A_{10}=\{14\}, A_{11}=\{15\}, A_{12}=\{17\}, A_{13}=\{19\}, A_{14}=\{21\}, A_{15}=\{22\}, \\
A_{16}=\{23\} .
\end{array}
$$

If $|A|>16$, then for $a, b \in A \quad a \neq b$, there must be $a, b \in A_{i_{0}}$ such that $ab$ is a perfect square; take $A=\left\{i ; i\right.$ is the largest number in $A_{i}$}, then $A$ satisfies the condition and $|A|=16$; if $|A|=16$, and $A A_{i}$ is a singleton set $\{i=1,2, \cdots, 16\}$, the number of such $A$ is $5 \times 3 \times 2 \times 2 \times 2=120$.