2000 AMC 12 Problems/Problem 23
Professor Gamble buys a lottery ticket, which requires that he pick six different integers from through , 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?
The product of the numbers have to be a power of in order to have an integer base ten logarithm. Thus all of the numbers must be in the form . Listing out such numbers from to , we find are the only such numbers. Immediately it should be noticed that there are a larger number of powers of than of . Since a number in the form of must have the same number of s and s in its factorization, we require larger powers of than we do of . To see this, for each number subtract the power of from the power of . This yields , and indeed the only non-positive terms are . Since there are only two zeros, the largest number that Professor Gamble could have picked would be .
Thus Gamble picks numbers which fit , with the first four having already been determined to be . The choices for the include and the choices for the include . Together these give four possible tickets, which makes Professor Gamble’s probability .
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