# Factors of 45: Prime Factorization, Methods, Tree, and Examples

**Factors**, an important category of statistical analysis, focus on breaking down a number (m) into a set of numbers that are capable of completely dividing m, without leaving any remainder behind.

In simple words, factors of a given number are the set of numbers that when divided by the number, result in a **whole-number quotient**, and give zero as the remainder.

For example,

The division leads us to a perfect whole-number quotient so, the number 1 is referred to as a factor of 45.

But,

\[ \dfrac {45}{2} = 22.5 \]

As the division fails to produce a perfect whole-number quotient, the number 2 is not considered as the factor of 45.

**Factors of 45 **are a collection of integers that, when **multiplied** together as a pair, result in 45 as the **product**. The numbers that are completely** divisible **by 45 are also referred to as its factors.

Like all the other sets of numbers, the factors of 45 are also categorized into **positive** and **negative **sets of integers. The only difference between the two sets is the **minus sign** that appears in front of the negative set of integers.

In the current article, we are going to put light on the methods and techniques used to calculate the factors of the number 45, its prime factorization, factor tree, and pairs of factors.

## What Are the Factors of 45?

**The factors of 45 are the follows: 1, 3, 5, 9, 15, and 45. Given that, these are the numbers, when multiplied in pairs, resulting in 45 as the product of their multiplication. **

The number 45 is a **composite number** by nature and has factors other than just a **universal factor**, i.e. **1** and **itself**. We can also say that the total number of factors of number 45 is **6**, as stated above.

## How To Calculate the Factors of 45?

You can calculate the factors of a given number (m) simply by using the commonly used **multiplication **or **division **methods as one of the primary techniques.

Here, as we are just focusing on finding the factors of 45, we will employ both the above-mentioned methods one at a time to construct a well-recognized list of the desired factors of 45.

To begin with, we will **multiply different pairs of numbers** to achieve the required result, of 45. In this way, the group of numbers that lead us to 45 as their product will be referred to as factors of the number 45.

**1 x 45 = 45 **

Similarly,

**3 x 15 = 45 **

**5 x 9 = 45 **

Hence,

**Factors of 45 = 1, 3, 5, 9, 15, 45 **

Now, we are going to find the set of factors of 45 by using the **division method**.

The division approach states that the recommended number (e.g. 1, 2, 3, 4, 5, 6,……, n) is to be considered a factor of 45 if it is divided by 45 and the division leaves no or zero remainders behind.

The procedures listed below should be used to calculate the factors of 45.

At first, we are going to divide the given number i.e. 45 by the smallest recommended number i.e. 1. Check for the remainder. Is the remainder zero?

\[ \dfrac {45}{1} = 45, r=0 \]

Yes, the remainder is zero.

Hence, we can say that the number 1 is a factor of 45.

Similarly, we are going to divide 45 by the number 2 such that,

\[ \dfrac {45}{2} = 22.5, r≠0 \]

No, the remainder is not equal to zero. Additionally, the division failed to yield a quotient of whole numbers.

Therefore, we can say that the number 2 is **not **a factor of 45.

Keep on dividing 45 by the other set of numbers using the same method, as described previously.

\[ \dfrac {45}{3} = 15 \]

\[ \dfrac {45}{5} = 9 \]

Hence,

**Factors of 45 = 1, 3, 5, 9, 15, 45 **

Each number has both positive and negative factors, as was already explained. Such that, the negative factors of the number are the **additive inverse** of its positive factors.

The following is the list of the negative factors of 45.

**Negative Factors of 45 = -1, -3, -5, -9, -15, -45 **

Similarly, the following is the list of the positive factors of 45.

**Positive Factors of 45 = 1, 3, 5, 9, 15, 45 **

## Factors of 45 by Prime Factorization

**Prime factorization **is the most widely used technique to find the **prime numbers** that when multiplied together, result in producing a whole number. The numbers that pair together to carry out the multiplication are termed the **prime factors**. Therefore, prime factorization is another method used to find the factors of any given number.

Now, to find the prime factors of a given number, a primary technique i.e. the prime factorization technique is used by following the unique **upside down-division** **methodology **commonly known as the **ladder method**.

The prime factorization of the number 45 is given as follows,

Also, the prime factorization of 45 can be expressed as the following expression,

**3 x 3 x 5 = 45 **

Hence, there are **3 **prime factors of 45.

**Prime Factors of 45 = 3, 3, 5 **

## Factor Tree of 45

A **factor tree **is the graphic representation of the prime factors of a number.

In the case of 45, the **prime numbers **3, 3, and 5 are considered to be its **prime factors**. Such that, the following image shows the factor tree of the number 45,

As seen from the image above, a factor tree, just like its visual representation shows prime factors of a number along its branches. Primarily where the tree finishes, the terminal branches are where the prime factors are displayed.

A few interesting facts about factors of the number 45 are as follows,

- The sum of the factors of 45 is (1+3+5+9+15+45) = 78.

- The factors of 45 are
**odd**, mainly due to the odd nature of 45.

**Factors of 45 = 1, 3, 5, 9, 15, 45 **

- Apart from the number 45 itself, the two composite numbers that are factors of 45 are 9 and 15, which are themselves the product of two prime numbers. Such that:

**3 x 3 = 9**

**3 x 5 = 15 **

## Factors of 45 in Pairs

The **pairs of factors** are those sets that consist of numbers that when multiplied by one another give the same number as the product of which they are a factor.

The factors of 45 are going to be called the **pair factors** when they are going to give the number 45 as the product of their **multiplication**. Luckily, the number 45 has **3** pairs of factors.

The pair of factors of the number 45 are represented as,

**1 x 45 = 45 **

Where, **(1, 45)** is a factor pair of 45.

Similarly,

**3 x 15 = 45 **

**5 x 9 = 45 **

Hence, **(3, 15), and (5, 9)** are the remaining factor pairs of 45.

The pair of factors can be both a set of **negative** or **positive** integers.

Hence, the positive factor pairs of the number 45 are given as,

** Positive Factor Pairs of 45 = (1, 45), (3, 15), (5, 9) **

Also, the negative factor pairs of 45 are given as,

**Negative Factor Pairs of 45 = (-1,-45), (-3, -15), (-5,-9) **

## Factors of 45 Solved Examples

Let us now solve a few examples to test our understanding of the above.

### Example 1

Windy wants to find the median of the factors of 45, such that, the number 45 is not included in the list. Can you help her in finding the correct answer?

### Solution

Given that:

The factors of 45 are given below:

** Factors of 45 = 1, 3, 5, 9, 15, 45 **

The factors of 45, excluding the number 45 from the list, is as follows:

** Factors of 45 = 1, 3, 5, 9, 15 **

A median is the central value of a list of factors.

By the above-mentioned data, **5** is the required value of the median.

### Example 2

Diana wants to calculate the common factors of the numbers 42 and 45. Can you help her in finding the desired C.Fs?

### Solution

The list of factors of 45 is given below:

**Factors of 45 = 1, 3, 5, 9, 15, 45 **

Also, the list of factors 42 is given below:

**Factors of 42 = 1, 2, 3, 6, 7, 14, 21, 42 **

The common factors of two numbers are those integers that co-exist as factors of both the proposed numbers.

Hence, the C.Fs of the numbers 42 and 45 are as follows:

**Common Factors = 1, 3**

The total number of common factors of 42 and 45 is** 2**, respectively.

### Example 3

Does Anne want to find the numbers between 1 to 9 which are not a factor of 45?

### Solution

The factors of 45 are given below:

**Factors of 45 = 1, 3, 5, 9, 15, 45 **

According to the above-mentioned list, the numbers between 1 to 9 which are not a factor of 45 are **2, 4, 6, 7, **and **8**.

*Images/mathematical drawings are created with GeoGebra. *