# 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). – Kentaro Tomono 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...................... – Kentaro Tomono May 31 '15 at 0:58

## 2 Answers

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!

• Thank you very much for your answer. Kindly let me add 1. I think I understand. If, I am afraid, you have time, I'm glad should you be able to help me regarding what I asked at the comment line. It might the tendency of the exam, although when students are asked, for example, "Do prime factoring this number etc etc", then the answer is always ( or it seems they should answer as simply as possible. What I mean here is, when prime-facoring, the students need to answer by the set of multiplication of prime numbers powered by "n". If you ask me an example, – Kentaro Tomono May 30 '15 at 19:43
• for example, when students are required to do prime factoring to number 288, the answer should be expressed as 284=2(2)*3(2)*23. ( Kindly be reminded () means a power ). This give me more pain in my head ( please kindly do not take literally though ), and actually more perplexes me. Is there such a rule in America's examination too?? ( = as simply as possible ). Thank you for your answer anyway first. --(m_m) – Kentaro Tomono May 30 '15 at 19:48
• Please kindly note what I commented is at Karl's answer's comment line. Sorry. – Kentaro Tomono May 30 '15 at 19:50
• That is one of the reasons why I asked this question. It seems, although may be that is the practice of my country's educational system, when using divisor, the number or polynominal should be expressed as much simply as possble. Kindly be noted that is why I asked this question. Anway thank you for your assistance. – Kentaro Tomono May 30 '15 at 19:57

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.

Having said that in terms of algebraic long division the expression you are dividing by is known as the divisor. In this sense there could possibly be a remainder.

• Thank you for your answer. Now, is there any rule in a case of prime factorization the answer should be discribed by 2 numbers? For example, here link it says Note: 12 = 2 × 2 × 3 can also be written using exponents as 12 = 2(2) × 3 and the answers of typical tests in my country are written always using exponents such as here link, for example 98=2×7(2). ( () is a power ) Thank you for your answer anyway though. – Kentaro Tomono May 30 '15 at 14:43
• You mean $2^3\times3$? I think this is still called a prime factor decomposition. – Karl May 30 '15 at 21:07
• If you highlight the maths you can see how to type using mathsjax – Karl May 30 '15 at 21:09
• I don't understand the American system either but the terminology is universal. You can factorise a number or write it as the product of its prime factors. – Karl May 31 '15 at 7:04
• @KentaroTomono Sometimes for emphasis we (Americans) say "factor completely," since in appropriate contexts writing a quantity as any sort of product can be called factoring. However, "prime factorization" always means a complete factorization into powers of primes. – user1527 May 31 '15 at 12:28