**Hint:**

If input string is in reverse order, you had better make use of the following property:
$$n^{\phi(m)}\equiv 1\pmod{m}, \quad \mathrm{if}\ (m,n)=1,$$
where $\phi(m)$ stands for the Euler's totient function.

In this specific case, $(10,7)=1$. Let the nodes of your DFA represent 2-tuples of current positions (mod $\phi(7)$ of course) and residue classes (mod $10$).

**More:**

In a general sense, it is easier to build a NFA that reads numbers from the most insignificant digit, since the base $n$ (in this case, $10$) and the divisor $m$ (in this case, $7$) do not necessarily coprime.