Maximum number of different decimal digits in $p/q$ for $p , q \in \Bbb Z$.

Question

In one of my class notes it was written that maximum number of different decimal digits in $p/q$ where $p , q \in \Bbb Z$ , is '$q$'. I don't know whether it is true or false, please tell me if it is correct or wrong with appropriate reason.

Answer 1

As stated the result is false since $\frac{246}{2}=123$ has $3$ digits.

For the number of digits after the decimal point, the result is true.

Since there are only $10$ digits we only need check $1\le q\le 9$. For $1,2,4,5,8$ the result is obvious.

The 'trickiest' case is $q=7$ but even without using a result such as Fermat's little theorem to prove your result neatly you only need check with a pocket calculator that each of $\frac{1}{7}, \frac{2}{7},..,\frac{6}{7}$ has precisely $6$ different digits.