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Abstract Algebra

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Euler's Function

ABSALG-1GWC3W

Let $m\ge 1$ be an integer and let $\varphi(m)$ be Euler's Phi function evaluated at $m$ (often written Euler's $\varphi$-function and also called Euler's totient function).

Which of the following statements is TRUE?

A

$a^{\varphi(m)}\equiv a\;{\rm mod}\; m$, for all integers $a$.

B

$a^{\varphi(m)}\equiv 1\;{\rm mod}\; m$, for all integers $a$.

C

Let $p>1$ be a prime number. Then $a^{p^2}\equiv a$ mod $p^2$ for all integers $a$ coprime to $p$.

D

The set of residues mod $12$ of the numbers $r=1,5,7,11$ equals the set of residues mod $12$ of the numbers $5r$, $r=1,5,7,11$.

E

The set of residues mod $12$ of the numbers $r=1,5,7,11$ equals the set of residues mod $12$ of the numbers $2r$, $r=1,5,7,11$.