Determine $x$ from the condition that the integer of the form $5^{x}$ is preceded by the number 7812500, which has no common divisor with $5^{x}$.
A number that is divisible by 5 and less than $5^{x}$ is $5^{x-1}$. The

$$
5^{x}-5^{x-1}=4 \cdot 5^{x-1}
$$

difference gives the number of numbers that do not share a common divisor with $5^{x}$. Thus:

$$
\begin{gathered}
4 \cdot 5^{x-1}=7812000 \\
5^{x-1}=1953125=5^{9}
\end{gathered}
$$

From this, it follows that

$$
x-1=9
$$

that is,

$$
x=10
$$