10. For the non-empty subsets of the set $A=\{1,2, \cdots, 10\}$, subsets whose elements' sum is a multiple of 10 are called good subsets. Therefore, the number of good subsets of $A$ is $\qquad$.
The subsets with a sum of 10 are $\{10\},\{1,9\},\{2,8\},\{3,7\},\{4,6\},\{1,2,7\},\{1,3,6\},\{1,4,5\},\{2,3,5\},\{1,2,3,4\}$, a total of 10;
The subsets with a sum of 20 are $\{9,8,3\},\{9,7,4\},\{9,6,5\},\{8,7,5\},\{3,4,5,8\},\{2,3,6,9\},\{2,4,5,9\},\{2,5,6,7\}$, $\{1,4,7,8\},\{1,5,6,8\}$, a total of 10;
And the combinations of two subsets with a sum of 10 are $9+4+4+2+2=21$;
For a sum of 30, adding 10 to the subsets with a sum of 20 (excluding 10) results in 22 subsets;
The subsets with a sum of 30 are $\{9,8,7,6\}$, a total of 1;
The combinations of two subsets with sums of 10 and 20 (excluding the element 10) are 14;
The combinations of three subsets with a sum of 10 (excluding the element 10) are 2;
The subsets with a sum of 15 are $\{5,10\},\{6,9\},\{7,8\},\{1,4,10\},\{1,5,9\},\{1,6,8\},\{2,3,10\},\{2,4,9\},\{2,5,8\}$, $\{2,6,7\},\{3,4,8\},\{3,5,7\},\{4,5,6\},\{1,2,3,9\},\{1,2,4,8\},\{1,2,5,7\},\{1,3,4,7\},\{1,3,5,6\}$, $\{2,3,4,6\},\{1,2,3,4,5\}$, a total of 20, so the subsets with a sum of 40 are 20;
The subsets with a sum of 50 are $A \backslash\{5\}, A \backslash\{1,4\}, A \backslash\{2,3\}$, a total of 3.
In summary, the good subsets of $A$ are 103 in total.