1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Numbers with sum = product
  1. Cross Product sum (Replies: 1)

Loading...