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Numbers with sum = product

  1. May 22, 2013 #1
    a + b + c = a * b * c

    where a, b, c are positive integers.

    I can think of only one solution to this. {1, 2, 3}.

    Is there any other solution to it?
    Can you prove or disprove?
     
  2. jcsd
  3. May 22, 2013 #2

    Office_Shredder

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    Without loss of generality assume [itex] a\geq b \geq c[/itex]. If [itex] b\geq c\geq 2[/itex] then
    [itex] a*b*c \geq 4a [/itex]
    Which is necessarily larger than [itex] a+b+c \leq 3a[/itex].

    So c=1 necessarily. Then we have
    [itex] a+b+1 = a*b[/itex]
    Now assume that b>2. The right hand side is at least 3a, and the left hand side is smaller than 2a+1, and we know that a is larger than 1 so these two cannot be equal. Therefore b=1 or b=2

    If b=1 and c=1 there is obviously no solution (we get a+2 = a). If c=1 and b=2 we get a+3 = 2a which is solved by a=3. So {1,2,3} is the only positive integer solution.
     
  4. May 23, 2013 #3
    Thank you.
     
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