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Math Help - Algebra

  1. #1
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    Algebra

    if a and b are integers and (a+b* sqrt(11) )> 1 and (a+b* sqrt(11) )* (a-b* sqrt(11) )=1, how do i show that a and b are positive integers? I was able to prove b>0 , but i can't seem to get the a >0 part, which makes me doubt whether i really showed b>0 .

    Thanks
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  2. #2
    Senior Member Sampras's Avatar
    Joined
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    Suppose  a+b\sqrt{11} > 1 . Also,  (a+b\sqrt{11})(a-b\sqrt{11}) = 1 . Let  c := a+b \sqrt{11} . Then  a-b \sqrt{11} = \frac{1}{c} . Then  ca-cb \sqrt{11} = 1 . Or  \sqrt{11} = - \frac{\frac{1}{c}-a}{b} . Or  \frac{1}{\sqrt{11}} = -\frac{b}{\frac{1}{c}-a} .

    We know that  0 < \frac{1}{c} < 1 . So  a > 0 . Also  b > 0 , which forces the quotient to be positive. If  b < 0 then  a < 0 \implies a+b \sqrt{11} < 1 . So  a >0 so that the quotient is negative.
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