Q:

Find all odd numbers $n$ such that $q= \frac{\ln(3n+1)}{6\ln(2)}$ is also an odd number. 

Find all odd numbers $n$ such that 
$$ q= \dfrac{\ln(3n+1)}{6\ln(2)}$$ is also an odd number. 

A:

The equation is equivalent to:
$$q=\log_{64}(3n+1)$$
or
$$n=\frac{64^q-1}3$$
Since $64\equiv 1\pmod 3$, $n$ is an integer.  And since $64^q-1$ is odd, $n$ is odd.  So you only have to substitute $q$ for odd numbers, and you get the possible values of $n$.

