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    Proof of the Irrationality using Euler's phi-function

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    2. Apply the proof of the irrationality of sqrt(2) to a) sqrt(3) and b)
    sqrt(4). If the proof breaks down, indicate precisely why.

    3. Euler's phi-function is defined such that for n > 0, phi(n) = |{m <
    n: gcd(m,n)=1}|. So, e.g., phi(4) = |{1,3}| = 2; phi(5) = |{1,2,3,4}| =

    4.
    a. Show that for prime p, phi(p) = p-1.

    b. Show that for prime p and q, phi(p*q) = (p-1)*(q-1).

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    2. Apply the proof of the irrationality of sqrt(2) to a) sqrt(3) and b)
    sqrt(4). If the proof breaks down, indicate precisely why.

    Solution:
    You haven't indicated which proof to use so I hope you now this one:
    a)Suppose
    is rational. This means that for some integers p and q
    Square both sides

    Multiply both sides by q2

    By Fundamental Theorem of Arithmetic, each of ...

    Solution Summary

    Irrationalities and primes are manipulated. The proof of the irrationality using Euler's phi-functions.

    $2.49

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