Sub-string divisibility

First, try with all multiples of 17 that has different digits:

$$\overline{cba} \equiv \mod 17$$

Since $a$, $b$ and $c$ are known values, the next step is:

$$\overline{dcb} \equiv \mod 13$$

to calculate the value of $d$:

$$100d + 10c + b \equiv 0 \mod 13$$ $$d \equiv -(10c+b)\text{inv}(100,13) \mod 13$$

More generally:

$$100z + 10y + x \equiv 0 \mod p$$ $$z \equiv -(10y+x)\text{inv}(100, p) \mod p$$

Since $x$ and $y$ are known values, $-(10y+x)\%p$ is also known. To calculate $\text{inv}(100, p)$ is enough with a preprocessing for every prime number from $2$ to $17$. Finally, the equation is similar to: $z \equiv k \mod p$, so for every digit the search space is reduced for just $1$ or $2$ values.

Time Complexity: $O(20\times 10\times 10)$

Just for $2$ and $5$ is needed to iterate from $0$ to $9$ and there are just $20$ numbers that met the requirements for $17$.