Decimal representation of $1/n$ includes the string of digits $777$ 
Find an integer $n$ with the property that the decimal representation of $1/n$ includes the string of digits $777$. 

For $\dfrac{m}{n}$ we can find a string of digits $777$ just by looking at $0.777$ and writing that as a fraction. But how do we do it for $1/n$?

$$\frac{7}{9}=0.777\ldots,\qquad \frac{7}{90}=0.0777\ldots $$
and if we consider
$$ \frac{7}{90000} = 0.0000777\ldots $$ we have
$$ \frac{7}{90000}+\frac{1}{10000} = 0.0001777\ldots $$
with
$$ \frac{7}{90000}+\frac{1}{10000}=\frac{7+9}{90000}=\frac{16}{90000}=\frac{1}{\color{red}{5625}} $$
since $7+9=16$ is a divisor of $9\cdot 10^4$. Even better: since $7+18$ is a divisor of $9\cdot 10^2$,
$$ 0.02\color{red}{777}\ldots=\frac{7}{900}+\frac{2}{100}=\frac{1}{\color{red}{36}}.$$
