How to solve this monster?

Question

I want to find $x$:

$$10^x\,\text{mod}\,17=0$$

Here, even wolframalpha refuses to answer. If there may be no solution, then how to prove that there is no solution?

EDIT: The equation below is equivalent:

$$10^x-17\Big\lfloor \frac{10^x}{17}\Big\rfloor=0$$

Answer 1

Since $10 = 2 \cdot 5$, then $10^x = 2^x \cdot 5^x$. Since 17 is prime, is it ever going to be the case that $17 \mid 2^x \cdot 5^x$?