2008 iTest Problems/Problem 91
Problem
Find the sum of all positive integers such that is satisfied by at least one ordered triplet of positive integers .
Solution (credit to official solution)
WLOG, let . We know that , so We also know that , so . Since are positive, . Thus, Note that if , then , so . Thus, , making . Now perform casework on the values of .
- If , then . Thus, must be a multiple of . The only value of that works is , so .
- If , then . Thus, must be a multiple of . The only value of greater than or equal to that works is , so .
- If , then . Thus, must be a multiple of . However, there are no values of greater than or equal to that work.
- If , then . Thus, must be a multiple of . However, there are no values of greater than or equal to that work.
The only values of that results in the equation solvable by positive integers is and , so the answer is .
See Also
2008 iTest (Problems) | ||
Preceded by: Problem 90 |
Followed by: Problem 92 | |
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