Hello, acc100jt!

A fascinating problem!

I have a game plan and a weak finish.

Well, maybe not . . .

Professor Gamble bought a lottery ticket; he picked six different integers from 1 to 46 inclusive.

He chose his numbers so that the sum of the base-ten logs 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?

Let his six numbers be: .

If the sum of the logs of his six numbers is an integer,

. . then the product of the 6 numbers is a power-of-10.

. . Hence: .

His product ranges from a minimum of: .

. . to a maximum of: .

Hence, his product is one of: .

. . We have only 7 cases to consider.

Can his product be ? . . . No

. . which canbe partitioned into 6 distinct factors.not

Can his product be ? . . . Yes

. . which factors into: .

Can his product be ? . . . Yes

. . which factors into: .

Can his product be ? . . . Yes

. . which factors into: .

Here's where it gets interesting.

Can his product be or larger? . . . No

The only numbers from 1 to 46 of the form .

. . are: .

The ones containing factors-of-5 are: .

Using all of them, we can account forfactors-of-5 . . . and no more.six

Hence. cannot be factored into six distinct factors from 1 to 46.

. . The same holds true for: .

I foundoutcomes that satisfy the log/sum/integer constraint.three

Therefore, the probability that Professor Gamble wins is