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Number Theory

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Moderate

Euler's Theorem: Moderate Statement Validity

NUMTH-WIB4XD

Euler's Theorem states that for integers $a$ and $m$ with $m\geq 1$,

$$ a^{\phi(m)}\equiv 1 \ (\mathrm{ mod } \ m),$$
for all $a$ coprime to $m$. Which of the following statements are true?

Select ALL that apply.

A

$\phi(m)|\phi(n)$ implies $m|n$.

B

$m|n$ implies $\phi(m)|\phi(n)$.

C

$\phi(m)=\phi(n)$ implies $m=n$.

D

If $a\geq 1$ and $a^{\phi(m)} \equiv 1 \ (\mathrm{ mod } \ m)$, then $m^{\phi(a)} \equiv 1 \ (\mathrm{ mod } \ a)$.

E

If $a\geq 1$ and $a^{\phi(a)} \equiv 1 \ (\mathrm{ mod } \ a)$, then $a=1$.