Problem:
Professor 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?
Answer Choices:
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Solution:
In order for the sum of the logarithms of six numbers to be an integer , the product of the numbers must be . The only prime factors of 10 are 2 and 5 , so the six integers must be chosen from the list , . For each of these, subtract the number of times that 5 occurs as a factor from the number of times 2 occurs as a factor. This yields the list , . Because has just as many factors of 2 as it has of 5 , the six chosen integers must correspond to six integers in the latter list that sum to 0 . Two of the numbers must be -1 and -2 , because there are only two zeros in the list, and no number greater than 2 can appear in the sum, which must therefore be . It follows that Professor Gamble chose , one number from , and one number from . There are four possible tickets Professor Gamble could have bought and only one is a winner, so the probability that Professor Gamble wins the lottery is .
OR
As before, the six integers must be chosen from the set , . The product of the smallest six numbers in is , so the product of the numbers on the ticket must be for some . On the other hand, there are only six factors of 5 available among the numbers in , so the product can only be , or .
Case . There is only one way to produce , since all six factors of 5 must be used and their product is already , leaving 1 as the other number: .
Case . To produce a product of we must use six numbers that include five factors of 5 and five factors of 2 among them. We cannot use both 20 and 40 , because these numbers combine to give five factors of 2 among them and the other four numbers would have to be odd (whereas there are only three odd numbers in ). If we omit 40 , we must include the other multiples of 5 plus two numbers whose product is 4 (necessarily 1 and 4. If we omit 20 , we must include , and 40 , plus two numbers with a product of 2 (necessarily 1 and 2 ).
Case . To produce a product of we must use six numbers that include four factors of 5 and four factors of 2 among them. So that there are only four factors of 2 , we must include , and 10 . These include two factors of 2 and four factors of 5 , so the sixth number must contain two factors of 2 and no 5 's, so must be 4 .
Thus there are four lottery tickets whose numbers have base-ten logarithms with an integer sum: , and . Professor Gamble has a probability of being a winner.
The problems on this page are the property of the MAA's American Mathematics Competitions