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Math Help - Probability contest problem

  1. #1
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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.
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  2. #2
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    Hello, Vicky!

    Professsor Gamble buys a lottery ticket, which requires that he pick six different integers from 1 -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?

    Suppose the winning numbers are: . a,b,c,d,e,f

    The sum of the six logs is an integer: . \log a + \log b + \log c + \log d + \log e + \log f \:=\:k

    This means that the log of the product of the numbers is an integer: . \log(abcde\!f) \:=\:k

    . . Hence: . abcde\!f \:=\:10^k

    That is, the product of the 6 different numbers is a power of 10.


    I found only four cases:

    . . \begin{array}{ccc}1\cdot2\cdot4\cdot5\cdot10\cdot2  5 &=& 10^4 \\<br />
1\cdot2\cdot5\cdot10\cdot25\cdot40 &=&10^5 \\<br />
1\cdot4\cdot5\cdot10\cdot20\cdot25 &=& 10^5 \\<br />
1\cdot5\cdot10\cdot20\cdot25\cdot40 &=& 10^6<br />
\end{array}


    Therefore, the Professor's probability of winning is \tfrac{1}{4}

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