Professor Gamble buys a lottery ticket, which requires that he pick six different integers from $1$ through $46$, inclusive. He chooses his numbers so that the sum of the base-ten [logarithms](https://artofproblemsolving.com/wiki/index.php/Logarithm) of his six numbers is an [integer](https://artofproblemsolving.com/wiki/index.php/Integer). It so happens that the integers on the winning ticket have the same property— the sum of the base-ten logarithms is an integer. What is the [probability](https://artofproblemsolving.com/wiki/index.php/Probability) that Professor Gamble holds the winning ticket?
$\textbf {(A)}\ 1/5 \qquad \textbf {(B)}\ 1/4 \qquad \textbf {(C)}\ 1/3 \qquad \textbf {(D)}\ 1/2 \qquad \textbf {(E)}\ 1$