2009 AIME I Problems/Problem 1
Call a -digit number geometric if it has distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers.
Assume that the largest geometric number starts with a . We know that the common ratio must be a rational of the form for some integer , because a whole number should be attained for the 3rd term as well. When , the number is . When , the number is . When , we get , but the integers must be distinct. By the same logic, the smallest geometric number is . The largest geometric number is and the smallest is . Thus the difference is .
Consider the three-digit number . If its digits form a geometric progression, we must have that , that is, .
The minimum and maximum geometric numbers occur when is minimized and maximized, respectively. The minimum occurs when ; letting and achieves this, so the smallest possible geometric number is 124.
For the maximum, we have that ; is maximized when is the greatest possible perfect square; this happens when , yielding . Thus, the largest possible geometric number is 964.
Our answer is thus .
The smallest geometric number is because and any number containing a zero does not work. is the largest geometric number because the middle digit cannot be 8 or 7. Subtracting the numbers gives
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