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

Assume that the largest geometric number starts with a nine. We know that the common ratio must be k/3, because a whole number should be attained for the 3rd term as well. When k = 1, the number is . When k = 2, the number is 964. When k = 3, 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 .