Contents

# Proper Factor|Definition & Meaning

## Definition

A **proper factor** of a number is **any factor** of that number other than **itself** or 1. For example, the factors of **24** are **1, 2, 3, 4, 6, 8, 12, and 24.** Of these, the **proper factors** are **2, 3, 4, 6, 8, and 12.**

**Factorization** is a common procedure used in **algebra** and other forms of **mathematics.** Any integer can be factorized into multiple **factors.** These factors have different **properties.** One type of factors is a **proper factor.** They are such **factors** that are not equal to 1 or the **original number** that was taken as an input to the **factorization process.Â **

The followingÂ **figure** illustrates an **example** of a proper factor that has been **iterated** in the **definition:**

**Figure 1: Example of Proper Factor**

## What Is Factorization and Proper Factor?

To understand the **process** of **factorization,** we need to understand **what factors are.** So let us see that in a bit more detail.

### Factor

**Factors** are **constituent** parts of an integer such that the **original** integer can be obtained by **multiplying** the factors **together.** For example, the **factors of 6 are 2 and 3**. If we multiply 2 and 3, we obtain **2×3 = 6,** which proves that 2 and 3 are the factors of 6 indeed. If you **divide** a number by itsÂ **factor**, the **remainder** is **zero**.

### Proper Factor

A **proper factor,** also known as a **proper**Â **divisor**Â in mathematics, is a factor of a number that is **greater than** one and less than the number itself that **evenly divides** the given number, **leaving zero** as a **remainder.** For example, let us say that we want to find the factors of the number 10.

Now let us **assume** for the moment that we know how to find these factors, and we find out that **10 can be factorized into 1, 2, 5, and 10**. Since 1 and 10 are not valuable, we say that **2 and 5 are the proper factors of 10**.

For all integers, two factors areÂ always there from the **definition** of factors. One of the factors is **1,** and the other one is the **number itself**. From the **definition** of factors stated above, we can see that these **numbers** are actually termed **factors,** but they **do not really add value** to our cause mathematically.

For example, if we say that **1 and 24 are the factors of 24**, I have just reiterated an obvious fact, and this information is **useless.** So, the proper factor is a type of factor that is not 1 or the **original number.**

Proper factors are a **fundamental** concept in mathematics, playing a key role in various areas of mathematics, including** number theory, algebra, geometry, and trigonometry**. Understanding and using **proper factors effectively** is essential for solving **mathematical problems** and making further advances in mathematics.

### Factorization

**Factorization** is the name for the **process** of **finding** the **factors** of an integer. Continuing from the previous **example,** we know that **2 and 3 are factors of 6**. But **how** did we arrive at these factors? How would I **find** these **factors** for a complex or difficult integer? To solve these **problems,** we need to understand the **factorization method.**

## Method of Finding Proper Factors

**Let us simplify**Â the **process** into steps. Let us say that we are asked to **find** the **factors** of an **integer** **X**. First we will create a **set of integers from 1 to X**.

Second, we will **divide X by all these integers** one by one. After every division, we will check whether the **remainder** of the division is **zero** or not. If the remainder is zero, then we store the **dividing number or the divisor** into another set since it’s a **factor** by definition.0

By the end of this process, we end up with a **set of factors**. Now to find the **proper factors,** we would **drop 1 and the original integer**Â and thus obtain the set of **proper fractions.** The step-wise **summary** is given below:

**Figure 2: Process of Finding Proper Factors**

**Step 1:** Define the **set of all potential factors** smaller than the given number excluding 1.

**Step 2:** **Divide the given number by all potential factors** one by one and note the remainder.

**Step 3:** Filter out the integers that have **zero remainders.**

### Example of Finding Proper Factors

Let us consider an **example.** Suppose we need to find the factors of the **number 14**. Now here are the steps:

(a) We define the **set of all integers smaller than 14, excluding 1,** which is given below:

**Potential Factors = { 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 }**

(b) Now we **divide 14 by all these numbers** one by one and note the remainder set given below:

**Remainders = { 0, 2, 2, 4, 2, 0, 6, 5, 4, 3, 2, 1 }**

(c) Now we **filter out the integers that have zero remainders** given below:

**Proper Factors = { 2, 7 }**

Hence, **2 and 7 are the proper factors of the number 14.**

**Figure 3: Example of Finding Proper Factors**

## Applications of Proper Factors

Proper factors are used in various **mathematical concepts** and **applications,** but for the sake of this article, let us consider the following four:

- Finding the
**greatest common divisor (GCD).** - Determining the
**prime factorization**of a number. - Solving problems in
**number theory.** - Solving problems in
**geometry and trigonometry.**

**Figure 4: Applications ofÂ Proper Factors**

Let us explain them one by one.

### 1. Finding the Greatest Common Divisor (GCD)

The **greatest common divisor** (also called the highest or greatest common factor) of two or more numbers is the **largest shared factor** between all of them. To find the **greatest common divisor,** one can **list** the **proper factors** of each number and then find the **largest one** that is common to all of them.

For example, the proper factors of **18 are 2, 3, 6, and 9,** and the proper factors of **24 are 2, 3, 4, 6, 8, 12**. As another example, the greatest common divisor of 24 and 18 is 6.

### 2. Determining the Prime Factorization of a Number

Another important use of proper factors is in finding the **prime factorization** of a number. The method of finding the prime factors of a given number that multiply together to form that **number** is called prime factorization.Â For example, the prime factorization of **24 is 2 x 2 x 2 x 3**.

The prime numbers are the only **proper** factors of a number that are also prime numbers, making it **easier** to perform **mathematical**Â **operations** with the number.

### 3. Solving Problems in Number Theory

In **number theory,** proper factors play a role in understanding **divisibility** and the **distribution** of **primes.** For example, it is known that the **quantity** of **proper factors** of a number is **directly related** to whether the number is **prime** or **composite.**

A prime number has only two**Â factors**, i.e., 1 and itself, whereas a composite number has more than two factors. This means that the prime numbers have no **proper factors.** This observation has been used to **develop algorithms** for finding prime numbers and to understand the **distribution** of **primes** in the number system.

**4. Solving Problems in Geometry and Trigonometry**

**Proper factors** are also used in solving problems in **geometry** and **trigonometry,** such as finding the **sides of** **a right triangle** when given the **area** and one side **length.** In such problems, **proper factors** can be used to **simplify expressions** and to determine the necessary **conditions** for a solution to exist.

## Numerical Problems of Proper Factors

**Example 1**

Calculate the **proper factors** of the **number 8**.

**Solution**

(a) We define the **set of all integers smaller than 8**, excluding 1, which is given below:

Potential Factors = { 2, 3, 4, 5, 6, 7 }

(b) Now we **divide 8 by all these numbers** one by one and note the **remainder set** given below:

Remainders = { 0, 2, 0, 3, 2, 1 }

(c) Now we filter out the** integers that have zero remainders** given below:

Proper Factors = { 2, 4 }

Hence, **2 and 4 are the proper factors of the number 8.**

**Example 2**

Show that the **number 11 is a prime number.**

**Solution**

(a) We define the **set of all integers smaller than 11,** excluding 1, which is given below:

Potential Factors = { 2, 3, 4, 5, 6, 7, 8, 9, 10 }

(b) Now we **divide 11 by all these numbers** one by one and note the **remainder set** given below:

Remainders = { 1, 2, 3, 1, 5, 4, 3, 2, 1 }

(c) Now we filter out the** integers that have zero remainders** given below:

Proper Factors = { }

This means that the **number 11 has no proper factors**. It is only divisible by 1 and itself. So **it’s a prime number.**

*All images were created with GeoGebra.*