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 |