If x+y+z=19,xyz=144 then find the value of x,y,z
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I don't have any great insight to offer, though I wouldn't be surprised if there is one to be had. I can offer what I think is a systematic way to generate all possible solutions, assuming that x y and z are positive integers as stated. The first step is to factor 144 completely
Then generate the set of possible factors of 144. I did this "by the seat of my pants," but I think the list is complete.
Obviously all numbers greater than 16 aren't useful for this problem. There are not 2 other integers you can add to 18, let alone 24, to get 19. That leaves
Now suppose . I then write and
From which I can get a quadratic equation
That has no real roots. This means and by symmetry neither does y or z; also, none of them x y or z can be 16 because then one of the others would have to be 1.
I haven't followed this procedure out to the bitter end, but I'm satisfied that the two solutions already given, namely and are the only solutions that exist.
BTW, it did occur to me to use combinations with repetition to find the number of integer solutions to x + y + z = 19, but I doubt that would've saved any time.