2007 iTest Problems/Problem 38
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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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