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Number Theory

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Properties of Euler Phi Function

NUMTH-T4KRE4

Denote by $\phi(n)$ the Euler totient function of a natural number $n$.

Which of the following statements are always true?

Check ALL that apply.

A

$\phi(n)=n-1$ for all $n$.

B

$\phi(mn)=\phi(m)\phi(n)$ for all $m$ and $n$.

C

Let $(m,n)$ be the greatest common divisor of $m$ and $n$, and $[m,n]$ be the least common multiple of $m$ and $n$. Then $\phi(m)\phi(n)=\phi((m,n))\phi([m,n])$.

D

$\phi(m+n)=\phi(m)+\phi(n)$ for all $m$ and $n$.

E

$\frac{\phi(n)}n = \frac{\phi(n^2)}{n^2}$ for all $n$.