anonymous
  • anonymous
if a and b are two integers with gcd(a,b)=1, then show that gcd(a-b, a^2+ab+b^2)=1 or 3
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
@experimentX
amistre64
  • amistre64
i wonder a-b = c a^2+ab+b^2 = d a-b+ab = c +ab a^2+ab+ab+b^2 = d +ab a-b+ab = c +ab (a+b)^2 = d +ab
anonymous
  • anonymous
then how to show gcd equals 1 or 3??????

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amistre64
  • amistre64
.... still looking at it :)
amistre64
  • amistre64
wondering if a euclidean algorithm would pan out a^2+ab+b^2 = (a-b)(q) + r ...
anonymous
  • anonymous
i am going to make a guess that it has something to do with \[a^3-b^2=(a-b)(a^2+ab+b^2)\] just a guess though
anonymous
  • anonymous
not a solution by any means, i just thought it might be the hook in to the problem i have no idea how to solve it
ParthKohli
  • ParthKohli
anonymous
  • anonymous
@ParthKohli thats it!!!
anonymous
  • anonymous
better if the whole thing is explained....
amistre64
  • amistre64
a^2+ab+b^2 = (a-b)(q) + r a^2+ab+b^2 - r= (a-b)(q) (a^2+ab+b^2 - r)(a-b) = q i was wondering if there was a method we could employ there; give that r=1 or 3, and gcd(a,b) = 1
Callisto
  • Callisto
*bookmark*
ParthKohli
  • ParthKohli
:-\ \[\dfrac{(a^2 + ab + b^2 - r)}{a - b} = q\]
ParthKohli
  • ParthKohli
Can anyone explain?
amistre64
  • amistre64
\[c|d~\iff~d=cn~;~n \in Z\] \[p = kq+r~\to \frac{p-r}{k}=q~,~q\in Z\]
amistre64
  • amistre64
the euclidean algorithm has alot of recurrsive:\[a=bq+r\] if \[a=bq+r\\b = r(b)+0\]then gcd(a,b)=r
amistre64
  • amistre64
hmmm, so if we can show that: a = r(b^2+1) or a/(b^2+1) = r it looks good to me :) is there a flaw in my thought perhaps? if not then: \[\frac{a^2+ab+b^2}{(a-b)^2+1}=1~or~3\]should do it :)
amistre64
  • amistre64
pffft, misread my own thoughts lol if b=rb then a = rbq+r a = r(bq+1) for q in Z

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