A number of programming languages construct large integers through 'concatenating' the digit to the end of the existing number. For example, Labyrinth, or Adapt. By concatenating the digit to the end, I mean that, if the existing number is \$45\$, and the digit is \$7\$, the result number is \$457\:(45 \times 10 + 7)\$.
A constructed number is a number that can be built this way through the use of the multiples of single digit numbers: \$1, 2, 3, 4, 5, 6, 7, 8, 9\$ A.K.A an element in one of these 9 sequences:
$$1, 12, 123, 1234, 12345, \: \dots$$ $$2, 24, 246, 2468, 24690, \: \dots$$ $$3, 36, 369, 3702, 37035, \: \dots$$ $$4, 48, 492, 4936, 49380, \: \dots$$ $$5, 60, 615, 6170, 61725, \: \dots$$ $$6, 72, 738, 7404, 74070, \: \dots$$ $$7, 84, 861, 8638, 86415, \: \dots$$ $$8, 96, 984, 9872, 98760, \: \dots$$ $$9, 108, 1107, 11106, 111105, \: \dots$$
To provide an example of how the sequences are constructed, here's how the sequence for \$a = 3\$ in constructed:
$$\begin{align} u_1 = && a = && 3 & = 3\\ u_2 = && 10 \times u_1 + 2 \times a = && 30 + 6 & = 36 \\ u_3 = && 10 \times u_2 + 3 \times a = && 360 + 9 & = 369 \\ u_4 = && 10 \times u_3 + 4 \times a = && 3690 + 12 & = 3702 \\ u_5 = && 10 \times u_4 + 5 \times a = && 37020 + 15 & = 37035 \\ u_6 = && 10 \times u_5 + 6 \times a = && 370350 + 18 & = 370368 \\ \end{align}$$ $$\vdots$$ $$\begin{align} u_{33} = && 10 \times u_{32} + 33 \times a = && 37\dots260 + 99 & = 37\dots359 \\ u_{34} = && 10 \times u_{33} + 34 \times a = && 37\dots359 + 102 & = 37\dots3692 \end{align}$$ $$\vdots$$
\$u_{33}\$ and \$u_{34}\$ included to demonstrate when \$n \times a \ge 100\$. A lot of digits dotted out for space.
It may still not be clear how these sequences are constructed, so here are two different ways to understand them:
Each sequence starts from the single digit. The next term is found by taking the next multiple of that digit, multiplying the previous term by \$10\$ and adding the multiple. In sequence terms:
$$u_n = 10 \times u_{n-1} + n \times a, \: \: u_1 = a$$
where \$a\$ is a single digit (\$1\$ through \$9\$)
Each of the \$9\$ elements at any point in the sequence (take \$n = 3\$ for instance) are the multiples of \$123\dots\$ from \$1\$ to \$9\$, where \$123\dots\$ is constructed by \$u_{n+1} = 10 \times u_n + n\$ \$(1, 12, 123, \dots, 123456789, 1234567900, 12345679011, \dots)\$
So the first values are \$1 \times 1, 2, 3, \dots, 8, 9\$, the second are \$12 \times 1, 2, 3, \dots, 8, 9\$, the third \$123 \times 1, 2, 3, \dots, 8, 9\$, etc.
Your task is to take a constructed number as input and to output the initial digit used to construct it. You can assume the input will always be a constructed number, and will be greater than \$0\$. It may be a single digit, which maps back to itself.
You may take input in any reasonable manner, including as a list of digits, as a string etc. It is acceptable (though not recommended) to take input in unary, or any other base of your choosing.
This is a code-golf so the shortest code wins!
Test cases
u_n => a
37035 => 3
6172839506165 => 5
5 => 5
246913580244 => 2
987654312 => 8
61728395061720 => 5
1111104 => 9
11111103 => 9
111111102 => 9
2469134 => 2
98760 => 8
8641975308641962 => 7
or as two lists:
[37035, 6172839506165, 5, 246913580244, 987654312, 61728395061720, 1111104, 11111103, 111111102, 2469134, 98760, 8641975308641962]
[3, 5, 5, 2, 8, 5, 9, 9, 9, 2, 8, 7]
When I posted this challenge, I didn't realise it could be simplified so much by the method used in Grimmy's answer, and so therefore would be very interested in answers that take a more mathematical approach to solving this, rather than a 'digit' trick (Obviously all valid answers are equally valid, just what I'd be interested in seeing).
