2009 AIME I Problems/Problem 1

Revision as of 21:14, 20 March 2009 by God of Math (talk | contribs) (Solution)

Problem

Call a $3$-digit number geometric if it has $3$ 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 $931$. When k = 2, the number is 964. When k = 3, we get $999$, but the integers must be distinct. By the same logic, the smallest geometric number is $124$. The largest geometric number is $964$ and the smallest is $124$. Thus the difference is $964 - 124 = \boxed{840}$.

See also

2009 AIME I (ProblemsAnswer KeyResources)
Preceded by
First Question
Followed by
Problem 2
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All AIME Problems and Solutions
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