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Question about irrational numbers

  1. Feb 7, 2010 #1
    Let p and q be distinct primes. Prove that [tex]\sqrt{p/q}[/tex] is a irrational number.
  2. jcsd
  3. Feb 7, 2010 #2
    It isn't a homework. I just need to prove it!

    Thank you,
  4. Feb 7, 2010 #3
    It's quite easy. Assume, that [tex] \sqrt{p/q}=a/b[/tex], where a and b are relative primes, ie GCD (a,b)=1.

    This is equivalent to [tex]pb^2=qa^2[/tex]. Since p and q are distinct primes, p | a^2 => p | a => The right side is divisible by p^2, and this is a contradiction, because the left side is not (because b is not divisible by p, since GCD (a,b)=1)
  5. Feb 7, 2010 #4
    I din't understand why b isn't divisible by p.

    Thank you for your answer!
  6. Feb 7, 2010 #5
    because if b is divisible by p, than GCD (a,b) is at least p, but we assumed that it equals to 1
  7. Feb 7, 2010 #6
    Thanks! :)
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