Greatest common divisor

  • Thread starter kathrynag
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  • #1
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Homework Statement


Suppsoe a, b[tex]\in[/tex]natural numbers, and d=GCD(a,b). Then d^2=GCD(a^2,b^2). I need to find where the proof goes wrong.


Homework Equations





The Attempt at a Solution


By hypothesis, we have that d divides a and d divides b, so there are integres s and t with a=ds and b=dt. Then a^2=d^2s^2 and so d^2 divides a^2. Similarily d^2 divides b^2. Thus d^2 is a common divisor of a^2 and b^2, as desired.
 

Answers and Replies

  • #2
HallsofIvy
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No, not "as desired". What you "desire" is the greatest common divisor. Saying that d^2 is a common divisor does not mean it is the GREATEST common divisor.
 
  • #3
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No, not "as desired". What you "desire" is the greatest common divisor. Saying that d^2 is a common divisor does not mean it is the GREATEST common divisor.

Ok, that made a lot more sense now that I see that!
 

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