text: 1. ## 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.
  
  2. 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: .$\displaystyle a,b,c,d,e,f$
  
  The sum of the six logs is an integer: .$\displaystyle \log a + \log b + \log c + \log d + \log e + \log f \:=\:k$
  
  This means that the log of the product of the numbers is an integer: .$\displaystyle \log(abcde\!f) \:=\:k$
  
  . . Hence: .$\displaystyle abcde\!f \:=\:10^k$
  
  That is, the product of the 6 different numbers is a power of 10.
  
  I found only four cases:
  
  . . $\displaystyle \begin{array}{ccc}1\cdot2\cdot4\cdot5\cdot10\cdot2 5 &=& 10^4 \\ 1\cdot2\cdot5\cdot10\cdot25\cdot40 &=&10^5 \\ 1\cdot4\cdot5\cdot10\cdot20\cdot25 &=& 10^5 \\ 1\cdot5\cdot10\cdot20\cdot25\cdot40 &=& 10^6 \end{array}$
  
  Therefore, the Professor's probability of winning is $\displaystyle \tfrac{1}{4}$
  
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