2000 AMC 12 Problems/Problem 23
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Problem
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?
Solution
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 of those 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 
.
See also
| 2000 AMC 12 (Problems • Resources) | ||
| Preceded by Problem 22 | Followed by Problem 24 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
