Contents

# Factor|Definition & Meaning

## Definition

A **number** that completely divides another number without leaving any **remainder** is called its **factor**. For example, 10 is **completely** divided by 2 and 5, and hence they are the **factors** of 10. Every **number** is **completely** divisible by itself and 1, so the **number** itself and 1 are always two of the various **factors** of a number.

## What Is a Factor?

**Factors **are a basic concept in **mathematics **that refer to the whole numbers that exactly **divide **another number. Let’s say we have the number 12, and the maximum possible factors of 12 can be 1, 2, 3, 4, 6, and the number 12 itself, as these **numbers **can be **multiplied **together to **equal **12.

There are some related **ideas **to factors, including prime numbers, greatest common factors (**GCF**), and least common multiples (**LCM**).

**Prime **numbers are unique numbers that can only be divisible by two numerals, i.e., 1 and the numbers themselves, while **GCF **represents the **largest **number that exactly **divides **two or more numbers. **LCM**, on the other hand, can be seen as the **most diminutive **number, which is only divisible by two or more than two **numbers**.

**Factors **have a **significant **impact in various areas of **mathematics**, including **arithmetic**, algebra, and **geometry**. In **arithmetic**, **factors **are utilized to determine the greatest common divisor (**GCD**) and **LCM **of two or more numbers. In **algebra**, they help simplify expressions and solve equations, while in **geometry**, they’re used to **calculate areas **and **volumes **of shapes.

**Factors **also play a crucial role in **number theory**, where researchers study the properties and patterns of numbers. For example, the distribution of **prime numbers **and the relationship between **prime **and **composite numbers **are studied in **number theory**.

Finding the **factors **of a **number **can be done by listing all the **numbers **that divide the **original **number evenly or by **dividing **the **number **by each whole number until there’s no longer a **remainder**.

If we **multiply** two numbers and it **produces** a whole number as a **product**, then the numbers being **multiplied** are known as **factors**. In other words, **factors** are also known as **divisors** as they entirely **divide** the numbers and yield **zero** remainders.

Factors provide **whole** numbers as **answers,** but they can be positive as well as **negative**. For example, **factors** of 20 are 1, 2, 4, 5, and 10 because these **numbers** factor 20 but -1, -2, -4, -5, and -10 are also **factors** of 20 as two **negative** numbers when **multiplied**, providing a **positive** number. But **generally**, only **positive** factors are considered.Â

### Properties of Factors

The following are some properties of factors:

- Every
**number**has one**factor**in common, i.e., 1. **Natural**numbers are their own**factors**.- More than two factors exist for each
**composite**number.Â - Factors can only be whole numbers or
**integers;**they can never be decimals or**fractions**. - 1 and the
**number**itself are the only factors of**prime**numbers.Â - Except for 0 and 1, every
**whole**number has, at the**minimum,**two factors. - All the
**even**numbers have one**typical**factor, i.e., 2. **Factors**are always**lesser**than or equal to the**assigned**number.- The number of factors can never be
**infinite;**it is always**finite**. - Factors can be
**estimated**by**operating**division and multiplication.

## Factors and Multiples

Since **factors** are the numbers being **multiplied**, multiples are the answers to these **multiplications**. In other words, **multiples** are the **product** of any two factors. Let us clear this concept using an **example;** we know that 2*3=6. Here, 2 and 3 are the **factors** of 6, and 6 is a multiple of 2 and 3. Moreover, **multiples** of 6 are 6,12,18,24,30,36, and so on.

**Multiples** can be noticed in **multiplication** tables. Unlike **factors**, Multiples are the **exceeding** values or equal **values** to the assigned number.Â

## How To Find Factors?

Now we must learn the **methods** of **obtaining** factors. One can easily and **quickly** estimate the factors if one has a good **command** over **multiplication** tables. The **simplest** methods of finding factors are **multiplication** and division. Let us discuss how to find factors using these two methods:

### Finding Factors Using Multiplication

We can estimate the factors by **operating** multiplication. First, we have to list down all the **numbers** that multiply to give the **assigned** number, for instance, 12. Now start making pairs of the two numbers like 1*12, 2*6, and 3*4. From 1 to 9, only these numbers **multiply** to give 12, so both of these **multiplicands,** i.e., 1,2,3,4,6 and 12, are the factors of 12.

### Finding Factors Using Division

Finding a single **factor** or multiple **factors** of a number is an easy method when using division. To estimate the factors using the **division** method, we have to list all the numbers that **wholly** divide the **assigned** number, let’s suppose the assigned **number** is 16. From 1 to 9, the numbers that give zero **remainders** are 16\1,16\2, 16\4, 16\8, and 16\16. So 1,2,4,8, and 16 are the factors of 16.Â

## Factor Tree

It is a method to **express** and find the **factors** of any number. The **numbers** that multiply to give a **particular** number can be written in a **tree-like** shape. It is an easy and simple **representation** of factors.

The tree stops where the **branches** reach the prime numbers, and the factors are **obtained**. It is to be noted that **1 **will not occur in the **factor **tree since it is not a **prime **number. Factor trees basically **portray **the prime factorization of numbers. Following is an **example **of a factor tree representation of 24.

So, we get 2*2*2*3 as the prime factor of the number 24. Or it can also be written as (2$^3$)*3.

## Types of FactorsÂ

Factors can be categorized as **common** factors, prime factors, and the highest common **factors**. Let us discuss these types of factors in detail.

### Common FactorsÂ

If two or more numbers **share** some similar **factors,** then we can call these factors **common** factors. For example, common factors of 24 and 36 are 1,2,3,4,6 and 12 because both 24 and 36 are **entirely** divided by these numbers. The **common **divisor is another name for common **factors**.

### Prime FactorsÂ

The **prime** numbers that **multiply** in order to produce an **integer** are going by the name of prime factors. The **method** of **evaluating** prime factors is called **prime** factorization. For example, the prime **factorization** of 12 gives 2*2*3, so 2 and 3 are the prime factors of 12.

## Highest Common Factor

As the name indicates, the **highest** common factor is the **largest** value among all the **common** factors of assigned numerals. To find the **highest** common factor, first, we have to find all the **common** factors of the **two** or more numbers. We discussed the common **factors** of 24 and 36. Here, 12 is the **largest** value among all the **common** factors, so it is the **HCF** of 24 and 36. HCF of 8 and 12 is shown in the figure below,

## Factors of Perfect Numbers

A **number** whose factors, when added, produce a number equivalent to itself is represented as the **perfect** number. The perfect number includes the **sum** of all the factors **except** the number itself. For example, 1, 2, and 3 are **factors** of 6. 1+2+3 = 6, so 6 is a **perfect** number.

Suppose **another** number, i.e., 10, factors of 10 are 1,2 and 5. 1+2+5 = 8; hence 10 is not a **perfect** number. The perfect **number** after 6 is 28. Every other **number** between 6 and 28 is not perfect. The number 28 has factors excluding itself, 1, 2, 4, 7, and 14. **Adding** them up, we get 28 as a result. Hence proved that 28 is a **perfect** number.

## Factors of Algebraic ExpressionÂ

As **natural** numbers have factors, algebraic **expressions** can also be expressed in the form of **factors**. Starting with a simple **example**, let 12x be an algebraic expression. The **factors** of 12x are 1, 2, 3, 4, 6, 12, x, and 12x.

Now talking about a **complex** algebraic **expression** and finding its factors, we take 11x$^2$ + 12xy – 3z$^2$ as an **algebraic** expression. This expression comprises 3 terms, i.e., 11x$^2$, 12xy, and -3z$^2$. The factors of these **individual** terms are:

11x$^2$ = 1.11.x.x

12xy = 1.2.2.3.x.y

-3z$^2$ = -1.3.z.z

Factors of algebraic **expressions** can also be determined by using common factors, using **identities**, and by **regrouping** the terms.

### By Using Common Factors

**Determining** the factors of algebraic **expression** by common factors, we understand this **concept** using an example as follows:

= 12xy$^2$ – 18yz$^2$

= 2.2.3.x.y.y – 2.3.3.y.z.z

= 2.3.y(2.x.y – 3.z.z)

= 6y(2xy – 3z$^2$)

### By Using Regrouping Method

There are some **expressions** that do not have any factors in **common**, so we have to use a different **method** for finding factors. We name it a **regrouping** method.

Let us suppose an **expression** 16x + y – xy – 16. There is no **common** factor among these four terms, but the first and **last** terms have a **common** value; **similarly**, the middle two terms share a **common** value. Hence we can group these terms using the following steps:

= 16x – 16 – xy +y

= 16(x-1)-y(x-1)

= (16-y)(x-1)

So we can **factorize** an algebraic **expression** by regrouping its terms.Â

### By Using IdentitiesÂ Â

In mathematics, there are some **well-known** identities or formulas which are used for **finding** the factors. These are as follows:

(a+b)$^2$ = a$^2$+2ab+b$^2$

(a-b)$^2$ = a$^2$-2ab+b$^2$

a$^2$-b$^2$ = (a+b)(a-b)

Let us take an **algebraic** expression, 16x$^2$ – 24xy + 9y$^2$. Factoring it using the identity as follows:

16x$^2$ – 24xy + 9y$^2$

=(4x)$^2$ – 2(4x)(3y) + (3y)$^2$

=(4x – 3y)$^2$

Hence, (4x – 3y)$^2$ is the factor of the expression 16x$^2$ – 24xy + 9y$^2$.

### Factor PairsÂ

When two **integers** multiply to give a certain **numeral** value, the **combination** of those **integers** is called factor pairs. In other words, it is a **pair **of two values that **multiply **to produce a product. There can be many **factor **pairs for a particular number except prime numbers. Since prime **numbers **have only 2 factors, so there can be only a **1-factor**** **pair for them.

For example, factors of 7 are 1 and **7, **so the factor pair for 7 is (1, 7). **Multiplication** tables make it easy to form **factor** pairs. For example, 2 and 12 are **factors** of 24, but they are not the only factors. 4 and 8 are also factors of 24. We can **write **it as (2, 12) and (4, 8) are the **factors **of 24.

Factor pairs can also be created for **negative **integers. Like (-2, -12) and (-4, -12) are also the **factors **of 24. Moreover, the factor pairs of **decimals**, fractions, and algebraic **expressions **can also be formed in a similar manner.

### Expanding and Factoring

**Expanding **is the opposite of factoring. Since **factoring **splits a number or an equation into factors,Â **expanding **multiplies factored values and returns back to the **actual **statement. Let us clarify this **concept **using an example.

Let 9y$^2$+24yz+16z$^2$ be an **algebraic **expression. Factors of this **expression **will be (3y+4z)$^2$ by using **identities**. (3y+4z)$^2$ is equal to (3y+4z)(3y+4z). Now if we want to **expand **it, we multiply both **factors **such that:

3y(3y+4z) + 4z(3y+4z) = 9y$^2$ + 12yz + 12yz + 16z$^2$ = 9y$^2$ + 24yz +16z$^2$

Thus, we proved that **expanding **and factoring are **inverse **of each other.

### Daily Life Factoring

In **daily **life practices, we often utilize **factors **without realizing them. Factors are **important **for various daily **grinds **like if you want to group or **divide **something, calculate and exchange money, units **converting** and time **distributing**, distributing workload, calculating your **traveling**, **factorization **in coding, and many more.

So we can **conclude **that one can easily become efficient in **dealing **with these problems if one has a piece of **great **knowledge and command over factors and **calculating **them.Â

## Solved Examples Involving Factor Calculations

### Example 1

Write down all the **factors** of 48 and 54. Also, find the highest **common** factor.

### SolutionÂ

Factors of **48** are:

1 * 48 = 48

2 * 24 = 48

**3 * 16 = 48**

4 * 12 = 48

6 * 8 = 48

Thus 48 has a total of 10 factors which are **{****1, 2, 3, 4, 6, 8, 12, 16, 24, 48****}.**

**Factors** of 54 are:

1 * 54 = 54

2 * 27 = 54

**3 * 18 = 54**

6 * 9 = 54

Thus 54 has a total of 8 factors which are **{1, 2, 3, 6, 9, 18, 27, 54}.**

Common factors of **1, 2, 3,** and 6. Out of these four factors, **6** is the highest common factor.

### Example 2

Write all the **positive** and negative factor pairs of 105.

### SolutionÂ

Positive **factor** pairs of 105:

1 * 105 = 105 Â (1, 105)

3 * 35 = 105 Â Â **(3, 35)**

5 * 21 = 105 Â Â (5, 21)

**7 * 15 = 105** Â Â (7, 15)

**Negative** factor pairs of 105:

-1 * -105 = 105 Â **(-1, -105)**

**-3 *** -35 = 105 Â Â (-3, -35)

-5 * -21 = 105 Â Â **(-5, -21)**

-7 * -15 = **105** Â Â (-7, -15)

*All images are created using GeoGebra.*