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Elementary theory of numbers and congruences
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HINT: First use Euler's criterion to make sure that they have solutions. Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely, Let $p$ be ...

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Solve the following equations for $x, (x ∈ N^+) (a)\ 2φ(x) = x; (b)\ 3φ(x) = x; (c)\ 4φ(x) = x.$
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$(a) 2φ(x)=x$ $x = 2^2$ $φ(x)=2^1*(2-1)=2$ $2φ(x)=2*2=2^2=x \Rightarrow 2φ(x)=x$ $(b) 3φ(x)=x$ $x=2*3^3=54$ $φ(x)=φ(2*3^3)=φ(2)φ(3^3)=18$ $3φ(x)=3*18=54=x \Rightarrow 3φ(x)=x$ $(c) 4φ(x)=x$ ...

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