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# Factors of 108: Prime Factorization, Methods, Tree, and Examples

In terms of its**Â extensive range of divisors**, the number 108 is particularly

**Â unique**Â in mathematics. Because 108 has so many factors, one result is that it is aÂ

**practical number**, which means that every positive integer

**Â less than 108Â**that is not a factor of 108 may be written as the sum of some of 108’s factors. According to the laws of multiplication,Â

**factors of 108**Â are numbers that when multiplied in pairs, give 108 as the product. Factors of a number can also be termed as itsÂ

**divisors**. Such that, the divisors of a number are theÂ

**set of integers**Â that when undergoing division, divide the numberÂ

**evenly**Â and leave no remainder behind.Â Â You may determine the factors of a given number using a variety of techniques, including division, multiplication, prime factorization, and factor tree. Fortunately, the present article will clarify each of the aforementioned approaches and demonstrate how to compute the factors of 108 using them.

## What Are the Factors of 108?

**The factors of 108 are the following: 1, 2, 3 , 4, 6, 9, 12, 18, 27, 36, 54, and 108.Â**All the aforementioned are the factors of the number 108 as these are the set of integers that when divided by the number 108, result in producingÂ

**zero**Â as the remainder.Â Â Due to itsÂ

**highly composite nature**, 108 is one of the most unusual positive numbers and has more factors than any other positive number smaller than 108.Â In simple words, the number 108 has a total ofÂ

**12**Â factors.

## How To Calculate the Factors of 108?

You can calculate the factors of 108 by using a variety of simple and useful techniques such as**divisionÂ**andÂ

**multiplication techniques,**Â which serve as the two main approaches.Â According to the laws of general division, numbers completely divide a given number so that a whole number is obtained as a quotient. Such numbers are referred to as the factors of the number. Letâ€™s use the same approach to calculate the factors of 108. Such that,Â

- Initially, divide a list ofÂ
**natural numbersÂ**that are less than and equal to the number 108.Â

**List of natural numbers = 1, 2, 3, 4, 5, 6, â€¦â€¦, 108.**

- Once the division process takes place, sort the numbersÂ
**completely divisibleÂ**by 108 from all the other numbers that generated aÂ**decimalÂ**or**Â fraction-based**Â product.Â - In the end, all theÂ
**perfect divisors**Â are going to be placed in a well-defined list known as the factors list of 108.Â Â

**multiplication**, by the previously employed methodology.Â Given below are the numbers that when multiplied together produce 108 as a result. The multiplication is given below:

**1 x 108 = 108**

**2 x 54 = 108**

**3 x 36 = 108**

**4 x 27 = 108**

**6 x 18 = 108**

**9 x 12 = 108**

**The numbers can have positive as well as negative factors**and each number’s negative factors. The negative factors are written with negative signs. The list of the positive factors of 108 is given as:

**Positive Factors of 108 :**

**1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108**Also, the list of the negative factors of 108 is given as:

**Negative Factors of 108:**

**-1, -2, -3, -4, -6, -9, -12, -18, -27, -36, -54, -108**Do you wish to learn some interesting facts regarding the factors of 108?Â

- Being claimed to beÂ
**dividedÂ**by theÂ**sum of its divisors**, the number 108 isÂ**refactorable**. - The list of 108 factors is distinctive since it combines bothÂ
**odd**Â andÂ**even**Â integers.

## Factors of 108 by Prime Factorization

Is there a simpler way for calculating the factors of 108 than the ones stated above? Yes!Â The most commonly used technique to split a number into its required factors is known asÂ**prime factorization**. Prime factorization is the simplest approach for finding a numberâ€™s prime factors. However, any number that is prime and can only be divided by themselves and by one is considered to be theÂ

**prime factor**Â of that number. The following steps are to be adopted while carrying out the prime factorization of 108: At first, divide the given number 108 by theÂ

**smallest possible prime number**Â i.e.Â Â \[ \dfrac {108}{2} = 54 \] Now, divide the quotient with the second smallest possible prime number. \[ \dfrac {54}{2} = 27 \] Continue to proceed until the only quotient left remains undividable. In other words, the following is the visual representation of how theÂ

**upside-down methodologyÂ**serves in evaluating the prime factors of 108.Â Â The prime factorization of 108 can also be expressed as the following expression:

**Prime factorization of 108 = 2 x 2 x 3 x 3 x 3**The

**Â**prime factors of 108 are as follows:

**Prime Factors of 108 = 2, 2, 3, 3, 3**

## Factor Tree of 108

AÂ**factor treeÂ**is a way of visually portraying the factors of a number, with each branch of the tree representing a factor of that number. The factor tree for the number 108 is depicted in the following figure:

## Factors of 108 in Pairs

Factors of a number when multiplied together in pairs such that, the result of the pair-multiplication is the originally recommended number, are known as theÂ**factor pairs**Â of that number.Â 108 being a uniqueÂ

**even composite number**Â leads to the generation of nearlyÂ

**6 positive pairs of factors**. Such that, each positive factor pair can also be written in the form of a negative pair of factors. Thus, considering both positive and negative integer pairings, there are a total ofÂ

**12-factor pairs**. The method for finding the factor pair of 108 is the same as the method for finding the factor pairs of any other integer. Therefore, the pair of factors for the number 108 are shown as:

**1 x 108 = 108**

**2 x 54 = 108**

**3 x 36 = 108**

**4 x 27 = 108**

**6 x 18 = 108**

**9 x 12 = 108**The positive factors in pairs of 108 are given as:

**Positive Factor Pairs of 108 = (1, 108), (2, 54), (3, 36), (4, 27), (6, 18), (9, 12)**The negative factors in the form of pairs can also be written similarly to positive factors are written. Therefore, theÂ

**negative**Â factor pairs of 108 are given as:

**-1 x -108 = 108**

**-2 x -54 = 108**

**-3 x -36 = 108**

**-4 x -27 = 108**

**-6 x -18 = 108**

**-9 x -12 = 108**The negative pairs of factors of 108 are given below:

**(-1, -108)**

**(-2, -54)**

**(-3, -36)**

**(-4, -27)**

**(-6, -18)**

**(-9, -12)**

## Factors of 108 Solved Examples

Now, let’s go through a few examples to test our understanding of the above article.### Example 1

Rina wants to calculate the average of the factors of 108. Help her to find the average.**Solution**The factors of 108 are given below: Factors of 108: 1, 2, 3 , 4, 6, 9, 12, 18, 27, 36, 54, and 108 Such that,Â The average of factors of 108 can be calculated by finding the sum of the above-mentioned factors 108 and dividing them by the total number of factors proposed in the list.

**Average = $\frac{\text{Sum of factors}}{\text{Total number of factors}}$**

**Average = $\frac{1+2+3+4+6+9+12+18+27+36+54+108}{12}$**

**Â Average = $\frac{280}{12}$**

**Average = 23.333Â**Hence, the average of the factors of 108 is 23.333, respectively.

### Example 2

Marry wants to find out which odd numbers, from 1 to 15, are not the factors of 108.Â### Solution

The factorsâ€™ list of 108 is given as:Â Factors of 108: 1, 2, 3 , 4, 6, 9, 12, 18, 27, 36, 54, and 108 Such that,Â From the aforementioned list, we can clearly say that the numbersÂ**5, 7, 11,Â**andÂ

**13**Â are the four odd numbers from 1 to 15 that are not factors of 108.

*Images/mathematical drawings are created with GeoGebra.Â*

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