2009 AIME I Problems/Problem 1

Problem

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.

Solution

Solution 1

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 .

Solution 2

Consider the three-digit number . If its digits form a geometric sequence, 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 .

Solution 3

The smallest geometric number is 124 because 123 and any number containing a zero does not work. 964 is the largest geometric number because the middle digit cannot be 8 or 7. Subtracting the numbers gives

See also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.