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!

Homework Help: Divisibility rules and proof by contradiction

  1. Jul 7, 2012 #1
    I posted this in the number theory forum to no success... so I figured maybe the homework help people would have some input

    Let x,y,z be integers with no common divisor satisfying a specific condition, which boils down to
    [itex]5|(x+y-z)[/itex] and [itex]2*5^{4}k=(x+y)(z-y)(z-x)((x+y)^2+(z-y)^2+(z-x)^2)[/itex]
    or equivalently [itex]5^{4}k=(x+y)(z-y)(z-x)((x+y-z)^2-xy+xz+yz)[/itex]
    I want to show that GCD(x,y,z)≠1, starting with the assumption 5 dividing (x+y), (z-y), or (z-x) results in x,y or z being divisible by 5. then it's easy to show that 5 divides another term, implying 5 divides all three.
    I run into trouble assuming 5 divides the latter part, [itex]2((x+y)^2+(z-y)^2+(z-x)^2)=((x+y-z)^2-xy+xz+yz)[/itex] and showing the contradiction from that point.
    Any hints?
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Jul 7, 2012 #2
    Hello tt2348,
    Leave the square terms unchanged and consider all possible values of the square of any integer mod 5 .There are three squares here() .How will you get them to get an expression divisible by 5 ?There lies the answer that one of the square terms has to be divisible by 5.
    Hoping this helps.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook