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:
For a -digit sequence to be geometric, there are numbers and such that the terms of the sequence are . The largest geometric number must have . Because both and must be digits less than must be a fraction less than with a denominator whose square divides . For , the largest such fraction is , and so the largest geometric number is . The smallest geometric number must have . Because both and must be digits greater than must be at least and so the smallest geometric number is . Thus the required difference is .
The problems on this page are the property of the MAA's American Mathematics Competitions