Is this statement true? $ \pi $ is a factor of $ 3 \pi $ but 3 isn't? What this question attempts to ask is, perhaps, the rigorous definition of factor itself.
I browsed the web for a definition of factor, and everywhere it was defined loosely (That common sense definition of factors).
But I have 3 questions -


*

*Can we extend the concept of factors to the realm of real numbers? like the one my original question seems to ask?

*Factors must necessarily divide a number perfectly (i.e., integrally) but the question is - must that factor also be an integer?

*if we agree that $\pi$ is a factor of $3\pi$ then that means every number has infinite number of factors like $\pi /2$, etc. Should we, then, redefine prime numbers?


Here is some rather elementary information which might be helpful.

The answer to your title-question is no, the statement is false.
Reasoning: Whenever you have a valid representation of a number as product of other numbers, then each of these numbers is called a factor.

Let's assume we are working with real numbers. A representation
\begin{align*}
3\cdot \pi
\end{align*}
is valid and we can conclude $3$ is a factor of $3\pi$ as well as $\pi$.

A few remarks to your questions:
  
  
*
  
*Ad (1) Can we extend the concept of factors to the realm of real numbers? 
  
  
  The answer is: Yes, we can. Whenever we have a multiplication $\cdot$, as we do have when working with reals, we can also talk about factors which are the constituents, the building blocks to do this multiplication.
  
  
*
  
*Ad (2) Factors must necessarily divide a number perfectly (i.e., integrally) but the question is - must that factor also be an integer?
  
  
  The formulation of this question is somewhat problematic. But first this answer: If there is a valid representation of real numbers given as product of factors, the factors need not be integers.

Example 1: The product $3\pi$ consists of two factors $3$ and $\pi$. One of them is an integer, the other is not an integer. No problem at all.
Example 2: The product $\pi \cdot \pi$ consists of two factors $\pi$ and $\pi$. None of them is an integer. No problem at all.
Note: The formulation divide a number perfectly (i.e., integrally) is problematic, since perfect is not a technical term in this context and integrally is not necessary in this context (see e.g. $\pi \cdot \pi$).

  
*
  
*Ad (3) if we agree that $\pi$ is a factor of $3\pi$ then that means every number has infinite number of factors like $\pi /2$, etc. Should we, then, redefine prime numbers? 
  
  
  This is really a clever question and it touches algebraic structures and number theory. It addresses questions like
  
  
*
  
*Which kind of numbers do we want to work with? Are these natural numbers, integers, reals or complex numbers or $\ldots$?
  
*What kind of multiplication is useful for these numbers? 
  
*If we have specified a certain kind of multiplication, what are the consequences. What are the factors, what is divisibility, which numbers of them should be called prime. 
The appropriate definition of a number being prime is part of algebra and will be defined there, when studying ring theory.

Final note: Sometimes you might work with and stick at  integers without using any  other numbers. In this specific context whenver you consider a product 
\begin{align*}
a\cdot b
\end{align*}
then $a$ and $b$ are integers and nothing else. In such cases a representation $3\cdot \pi$ is not valid and not under consideration, simply because $\pi$ is not an integer. The question is $\pi$ a factor? is in this context meaningless.
