TITLE: Explain why the following is or isn't a multiplicative function.
QUESTION [7 upvotes]: I'm working through practice problems for my exam in class and I need some help with the following problem:

A function $f(n)$ is defined to be the greatest power of $2$ that divides $n$.
For example, $f(20) = 2^2 = 4$, $f(32) = 2^5 = 32$, $f(72) = 2^3 = 8$, etc.
Is $f(n)$ a multiplicative function? Explain your answer.

I believe that it is a multiplicative function by looking at the examples, but I'm not sure how to explain the logic behind it and prove it for all cases.
Please correct me if I'm wrong, thank you for the help!

REPLY [3 votes]: Suppose $a = 2^\alpha p$ where $p$ is an odd prime, and $b = 2^\beta q$, where $q$ is also an odd prime (it may even be the same as $p$, actually). Thus $f(a) = 2^\alpha$ and $f(b) = 2^\beta$. Then what is $ab$? Obviously $ab = 2^\alpha 2^\beta pq = 2^{\alpha + \beta} pq$, so $f(ab) = 2^{\alpha + \beta}$. Now you only need to consider $p$ and $q$ that may not be prime but are odd to complete the proof.