# Does “factor” mean simply the multiplication (of any functions, numbers etc)

I am sorry I am not directed with the education of math.

But granted, let me ask the above question.

In my language (actually Japanese), the words corresponding with the factor and divisor, seem to have come from English. (Not confirmed.)

Now when teaching (potentially for now to me now) them to kids.

I am sorry to say my question is just as it is at the title.

Does factor simply mean the multiplication of any functions, or numbers, whereas, the divisor means literally and only "divide" any numbers, etc?

I am afraid I am a little confused about the relationship between these two terms.

Let me kindly take number 24. The number 24 can be expressed by 24=2*2*2*3 if it is expressed by the divisor(s). Now when considering the same 24, when expressed by the factor, 24=4*6 or either 2*12.

Am I correct about understanding the meaning and the difference of these 2 terms??

Kindly let me thank you in advance.

• Thank you for edting (m_m). – user5245 May 30 '15 at 14:45
• You are welcome. And welcome to the site! – Joonas Ilmavirta May 30 '15 at 20:02
• Why is there a vote to close? I think the question of differentiating between factors and divisors is good; I know that this is an issue in my class for elementary school teachers, since textbooks vary depending on the context (specifically in questions about factorization within elementary number theory vs. expressions of the form $a \div b$ in which $a, b \in \mathbb{Q}$). – Benjamin Dickman May 30 '15 at 21:41
• I am sorry for all of you by bomberding many comments, looks like I get lost. Even after celeriko's kind explanation, I still smell there might be something more about the word factor or factoring as I am saying at comment line of the answerers...................... – user5245 May 31 '15 at 0:58

From your examples, it seems that you generally understand the two terms and their differences/similarities. as @Karl says, they are more or less interchangeable for most audiences. Below I have elaborated a little bit more on the minute distinctions.

Unfortuneately, both of these words have two different meanings each that are subtle, yet distinct. I fear this is where your confusion might be stemming from

In general terms, a divisor is simply a number that divides another number, whether with remainder or without. In the following two expressions, $7$ would be the divisor even though the result is not an integer.

$$\frac{22}{7} \\ 22 \div 7$$

In addition, since $2 \cdot 3 = 6$, this also means that $2$ and $3$ are both divisors of $6$, in that they divide it without remainders, even though there isn't actually division happening, persay. A synonymous term for this type of divisor is a factor.

Factor can also used in two different ways 1) as a noun (object), which is like the $2 \cdot 3 = 6$ example above, and 2) as a verb (action). In 2) factor means the action of breaking a number/expression into is composite parts, such as taking the polynomial expression $x^2 + 7x + 12$ and writing it as the equivalent $(x+3)(x+4)$ where $(x+3)$ and $(x+4)$ are also factors of the original polynomial.

As you can see they are more or less the same however it may be worth it to show your students the difference if there are confusions that are arrising, i hope this helps!

The terms are interchangeable as I understand. Factors are divisors. $2\times2\times2\times3$ is a product of prime factors. The $8$ and $3$ are merely factors.
• You mean $2^3\times3$? I think this is still called a prime factor decomposition. – Karl May 30 '15 at 21:07