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 of his six numbers is an 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 that Professor Gamble holds the winning ticket? Provide your answer in the form of a simplified fraction $\frac{k}{m}$, and find the value of $k + m$.

Let's think step by step and output the final answer within \boxed{}.
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