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Probability contest problem

Professsor 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 winner ticket?

The answer is 1/4 and I got it after an hour of guessing and checking.

I summarized how I reached the answer, but I wanted to check if it is reasonable. I'm starting to see my solutions are too sloppy, and they could be wrong even though I got the correct answer. Could someone show me a way to solve this with less guessing?

Vicky.