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Number Theory

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Arithmetic Function with Special Properties: Cardinality

NUMTH-KE6KWF

Find the cardinality of the set of all arithmetic functions satisfying all of the following properties:

(1) For any $n\in \mathbb{N}$, $f(n)^{n}=1$.

(2) For any $n\in \mathbb{N}$, $f(n)^k \neq 1$ if $0< k < n$.

(3) If $m|n$, then $f(n)^{\frac nm}=f(m)$.

A

Finite.

B

$\aleph_0$. (Cardinality of the set of natural numbers)

C

$2^{\aleph_0}$. (Cardinality of the set of real numbers)

D

$2^{2^{\aleph_0}}$. (Cardinality of the power set of the set of real numbers)

E

$2^{2^{2^{\aleph_0}}}$. (Cardinality of the power set of the power set of real numbers)