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
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
.
See also
2009 AIME I (Problems • Answer Key • Resources) | ||
Preceded by First Question |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |