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Number Theory

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Euler's Theorem: Easy Statement Validity

NUMTH-FJJ3P4

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

$\phi(m)|m-1$ for any $m\geq 1$.

B

$a^{m-1}\equiv 1 \ (\mathrm{ mod } \ m)$ for any coprime $a$, $m$.

C

If $a^{\phi(m)} \not\equiv 1 \ (\mathrm{ mod } \ m)$, then $a$ and $m$ are not coprime.

D

If $a^{\phi(m)} \not\equiv 1 \ (\mathrm{ mod } \ m)$, then $a$ divides $m$.

E

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