Not as hard as it seems

Firstly, I eliminated possibilities for each digit of $\overline{abcdefghij}$ :

$j=0 \Rightarrow e=5$

$(b,d,f,h) \in (2,4,6,8) \Rightarrow (a,c,g,i) \in (1,3,7,9)$

$4|\overline{abcd} \Rightarrow 4|\overline{cd}$ , so $c \in (1,3,7,9) \Rightarrow d \in (2,6)$

$8|\overline{abcdefgh} \Rightarrow 8|\overline{fgh}$ , so $f \in (2,4,6,8), g \in (1,3,7,9) \Rightarrow h \in (2,6)$

Finally, $(d,h) \in (2,6) \Rightarrow (b,f) \in (4,8)$ .

Then, I made lists of each $n$ -digit number which was valid:

$\overline{ab} \in (14, 18, 34, 38, 74, 78, 94, 98)$

$\overline{abc} \in (147, 183, 189, 381, 387, 741, 783, 789, 981, 987$

I'll skip a few lines, because $d$ just tacks $(2,6)$ onto each and $e$ sticks on a $5$ , so each set has $20$ elements apiece.

$\overline{abcdef} \in (147258, 183654, 189654, 381654, 387654, 741258, 783654, \\789654, 981654, 987654)$

$\overline{abcdefg} \in (1472583, 3816547, 7836549)$

$\overline{abcdefgh} = 38165472 \Rightarrow \overline{abcdefghij} = 3816547290$