2007 iTest Problems/Problem 38
Problem
Find the largest positive integer that is equal to the cube of the sum of its digits.
Solution
Clearly this number is the cube of a positive integer so let be the number we seek. The sum of the digits of is equal to , so or for some natural . Because there is only one factor of 3 in a set of 3 consecutive numbers, one of , , or must be divisible by 9. Now note that has maximum 5 digits because else the prompt is always false (solution to ), so . Thus could be . The first three numbers are clearly ineligible because they would require too many high digits. Then we test each number. , , , , and finally , so the answer is .
See Also
2007 iTest (Problems, Answer Key) | ||
Preceded by: Problem 37 |
Followed by: Problem 39 | |
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