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!

Integers: 1,2,3,4,5,6,7,8,9

  1. Oct 19, 2009 #1
    I saw this somewhere, it looks like fun but i cant seem to answer it

    integers: 1,2,3,4,5,6,7,8,9

    [tex]\frac{a}{bc}+\frac{d}{ef}+\frac{g}{hi}=1[/tex]

    what is a,b,c,d,e,f,g,h,i ?
    pick from the above integers. (ONLY USE EACH OF THE ABOVE INTEGERS ONCE)
    :)

    bc means for example 35 (b=3 and c=5), not multiply.
     
  2. jcsd
  3. Oct 20, 2009 #2
    Re: fractions

    Interesting.























    We can always reorder terms such that b < e < h. Once we do that, it's easy to see that b=1 (because if b>1, the whole sum is necessarily less than 1).

    We can restrict the number of possibilities for a & c further, because, since b=1, d/ef+g/hi is at most 9/24+8/35 ~ 0.603 and therefore a/bc > 0.397. Furthermore, if c is 2, d/ef+g/hi <= 9/35+8/46 ~ 0.431 and a/bc > 0.569. If either a or c is 9, d/ef+g/hi <= 8/24+7/35 ~ 0.533 and a/bc > 0.467. If a and c are 8 and 9, d/ef+g/hi <= 7/24+6/35=0.463 and a/bc > 0.537 (so, a and c can't be 8 and 9, because both 8/19 and 9/18 are less than 0.537).

    c, f, and i can't be 5.

    Any two-digit prime number that occurs in decompositions of denominators must occur in at least two denominators. For example, we can rewrite the statement as (d*hi+g*ef)-ef*hi = -a*ef*hi/bc. Right hand side is integer. If bc is a multiple of 13, either ef or hi (or both) must also be multiples of 13.

    These principles limit the set of possibilities for the first fraction to 16 possibilities:

    7/12 8/12 9/12
    6/13 7/13 8/13 9/13
    6/14 7/14 8/14 9/14
    7/16 8/16 9/16
    8/17 9/17

    At this point, I don't see any clear ways to reduce the set of possibilities further, but we've cut the possibility space enough to start brute-force search for a solution ...
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Integers: 1,2,3,4,5,6,7,8,9
  1. 1-2+3-4+5 = 1/4 (Replies: 2)

  2. 6÷2(1+2) = 1 or 9? (Replies: 4)

Loading...