Problem 207

By the Triangle Inequality and applying the well-known logarithmic property $\log_{c} a + \log_{c} b = \log_{c} ab$, we have that

$\log_{10} 12 + \log_{10} n > \log_{10} 75$
$\log_{10} 12n > \log_{10} 75$
$12n > 75$
$n > \frac{75}{12} = \frac{25}{4} = 6.25$

Also,

$\log_{10} 12 + \log_{10} 75 > \log_{10} n$
$\log_{10} 12\cdot75 > \log_{10} n$
$n < 900$

Combining these two inequalities:
$$6.25 < n < 900$$
Thus $n$ is in the set $(6.25 , 900)$; the number of positive integer $n$ which satisfies this requirement is $\boxed{893}$.