One, (15 points) Given $T=\{1,2, \cdots, 8\}$. For $A \subseteq T, A \neq \varnothing$, define $S(A)$ as the sum of all elements in $A$. Question: How many non-empty subsets $A$ of $T$ are there such that $S(A)$ is a multiple of 3 but not a multiple of 5?
(Chen Yonggao provided)
For $\varnothing$, define $S(\varnothing)=0$. Let
$$
T_{0}=\{3,6\}, T_{1}=\{1,4,7\}, T_{2}=\{2,5,8\} \text {. }
$$

For $A \subseteq \dot{T}$, let
$$
A_{0}=A \cap T_{0}, A_{1}=A \cap T_{1}, A_{2}=A \cap T_{2} .
$$

Then $S(A)=S\left(A_{0}\right)+S\left(A_{1}\right)+S\left(A_{2}\right)$
$$
\equiv\left|A_{1}\right|-\left|A_{2}\right|(\bmod 3) \text {. }
$$

Therefore, $3|S(A) \Leftrightarrow| A_{1}|\equiv| A_{2} \mid(\bmod 3)$. There are the following cases:
$$
\begin{array}{l}
\left\{\begin{array} { l } 
{ | A _ { 1 } | = 0 , } \\
{ | A _ { 2 } | = 0 ; }
\end{array} \quad \left\{\begin{array} { l } 
{ | A _ { 1 } | = 0 , } \\
{ | A _ { 2 } | = 3 ; }
\end{array} \left\{\begin{array}{l}
\left|A_{1}\right|=3, \\
\left|A_{2}\right|=0 ;
\end{array}\right.\right.\right. \\
\left\{\begin{array} { l } 
{ | A _ { 1 } | = 3 , } \\
{ | A _ { 2 } | = 3 ; }
\end{array} \left\{\begin{array} { l } 
{ | A _ { 1 } | = 1 , } \\
{ | A _ { 2 } | = 1 ; }
\end{array} \left\{\begin{array}{l}
\left|A_{1}\right|=2, \\
\left|A_{2}\right|=2 .
\end{array}\right.\right.\right.
\end{array}
$$

Thus, the number of non-empty subsets $A$ that satisfy $3 \mid S(A)$ is
$$
2^{2}\left(\mathrm{C}_{3}^{0} \mathrm{C}_{3}^{0}+\mathrm{C}_{3}^{0} \mathrm{C}_{3}^{3}+\mathrm{C}_{3}^{3} \mathrm{C}_{3}^{0}+\mathrm{C}_{3}^{3} \mathrm{C}_{3}^{3}+\mathrm{C}_{3}^{1} \mathrm{C}_{3}^{1}+\right.
$$
$\left.C_{3}^{2} C_{3}^{2}\right)-1=87$.
If $3|S(A), 5| S(A)$, then $15 \mid S(A)$.
By $S(T)=36$, we know that the possible values of $S(A)$ that satisfy $3|S(A) 、 5| S(A)$ are $15 、 30$. And
$$
\begin{array}{l}
15=8+7=8+6+1=8+5+2 \\
=8+4+3=8+4+2+1=7+6+2 \\
=7+5+3=7+5+2+1=7+4+3+1 \\
=6+5+4=6+5+3+1=6+4+3+2 \\
=5+4+3+2+1, \\
36-30=6=5+1=4+2=3+2+1 .
\end{array}
$$

Therefore, the number of $A$ that satisfy $3|S(A) 、 5| S(A) 、 A \neq \varnothing$ is 17.
Thus, the number of $A$ that satisfies the conditions is $87-17=70$.