An algebra problem by mridul jain
Algebra
Level
3
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.
The answer is 840.
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1 solution
Assume that the largest geometric number starts with a nine. We know that the common ratio must be a rational of the form k/3 for some integer k, 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 = 840.