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 nine. 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 .
Maybe an easier way how to see the solution: Consider a three-digit number . If it is geometric, then we must have , or equivalently .
For we get , which is not an integer. Similarly, for we will get a non-integer . For we get , hence is the largest three-digit geometric number. And as obviously the smallest possible pair provides the solution , the answer is .
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