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Number Theory

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Definition of Multiplicative Function

NUMTH-G3JDKN

An arithmetic function $f$ is multiplicative if $f(mn)=f(m)f(n)$ for any coprime $m$ and $n$.

Which of the following functions are multiplicative?

Select ALL that apply.

A

$f(n)$ is the number of all positive integers at most $\cfrac{n^2}2$.

B

$f(n)$ is the number of all positive divisors of $n^2$.

C

$f(n)$ is the number of all prime numbers up to $n$.

D

$f(n)$ is the number of all positive integers at most $n^2$.

E

$f(n)$ is the number of distinct real roots of the equation $x^2 - 4nx +4=0$.