Find 7 positive integers
Find seven positive integers such that
Type your answer as the sum of these seven positive integers. If you think that there is no correct answer, type , or there are more than one distinct solution, type .
Clarification: Reordering the sequence of those integers does not count as another solution.
The answer is 10.
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1 solution
The only set that works (up to reordering) is { 1 , 1 , 1 , 1 , 1 , 2 , 3 } .
Without loss of generality, say x 1 ≤ x 2 ≤ ⋯ ≤ x 7 .
The right hand side of the equation is at most 1 4 x 7 2 .
From the left hand side, this means x 1 x 2 x 3 x 4 x 5 x 6 ≤ 3 , which only leaves three possible sets to check: we must have x 1 = x 2 = x 3 = x 4 = x 5 = 1 , then x 6 is 1 , 2 or 3 .
As it happens, only the one set above works.