1989 AIME Problems/Problem 14
Problem
Given a positive integer , it can be shown that every complex number of the form , where and are integers, can be uniquely expressed in the base using the integers as digits. That is, the equation
is true for a unique choice of non-negative integer and digits chosen from the set , with . We write
to denote the base expansion of . There are only finitely many integers that have four-digit expansions
Find the sum of all such .
Solution
First, we find the first three powers of :
So we need to solve the diophantine equation .
The minimum the left hand side can go is -54, so , so we try cases:
- Case 1:
- The only solution to that is .
- Case 2:
- The only solution to that is .
- Case 3:
- cannot be 0, or else we do not have a four digit number.
So we have the four digit integers and , and we need to find the sum of all integers that can be expressed by one of those.
:
We plug the first three digits into base 10 to get . The sum of the integers in that form is .
:
We plug the first three digits into base 10 to get . The sum of the integers in that form is . The answer is .
See also
1989 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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