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

In terms of its extensive range of divisors, the number 108 is particularly unique in mathematics. Because 108 has so many factors, one result is that it is a practical number, which means that every positive integer less than 108 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, factors of 108 are numbers that when multiplied in pairs, give 108 as the product. Factors of a number can also be termed as its divisors. Such that, the divisors of a number are the set of integers that when undergoing division, divide the number evenly and leave no remainder behind.    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.  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 producing zero as the remainder.   Due to its highly composite nature, 108 is one of the most unusual positive numbers and has more factors than any other positive number smaller than 108.  In simple words, the number 108 has a total of 12 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 division and multiplication techniques, which serve as the two main approaches.  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. Let’s use the same approach to calculate the factors of 108. Such that, 

1. Initially, divide a list of natural numbers that are less than and equal to the number 108. 

List of natural numbers = 1, 2, 3, 4, 5, 6, ……, 108.

1. Once the division process takes place, sort the numbers completely divisible by 108 from all the other numbers that generated a decimal or fraction-based product. 
2. In the end, all the perfect divisors are going to be placed in a well-defined list known as the factors list of 108.  

Now, begin by applying the division technique to find the smallest possible factor of 108. At first, divide the smallest possible divisible number i.e. 1 by the given number 108. Check for the remainder. Is the remainder zero? For the above division, the remainder is zero hence 1 is the factor of 108. The number 1 is also called a universal factor because it is divisible by every natural number. Continue to divide various numbers to determine whether or not they are factors of 108. \[ \dfrac {108}{2} = 54 \] \[ \dfrac {108}{3} = 36 \] \[ \dfrac {108}{4} = 27 \] \[ \dfrac {108}{6} = 18 \] \[ \dfrac {108}{9} = 12 \] Hence, the numbers 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108 are the factors of 108.  All numbers, other than those already listed, will not divide 108 evenly and leave some remainder. In simple words, numbers other than 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108 are not considered factors of 108.  To compute the factors of 108, we will now discuss the other widely used method, multiplication, by the previously employed methodology.  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? 

1. Being claimed to be divided by the sum of its divisors, the number 108 is refactorable.
2. The list of 108 factors is distinctive since it combines both odd and even integers.

Factors of 108 by Prime Factorization

Is there a simpler way for calculating the factors of 108 than the ones stated above? Yes!  The most commonly used technique to split a number into its required factors is known as prime factorization. Prime factorization is the simplest approach for finding a number’s prime factors. However, any number that is prime and can only be divided by themselves and by one is considered to be the prime factor 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 the smallest possible prime number i.e.   \[ \dfrac {108}{2} = 54 \] 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 the upside-down methodology serves in evaluating the prime factors of 108.    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 prime factors of 108 are as follows: Prime Factors of 108 = 2, 2, 3, 3, 3

Factor Tree of 108

A factor tree 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 the factor pairs of that number.  108 being a unique even composite number leads to the generation of nearly 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 of 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, the negative 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,  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 = $\frac{1+2+3+4+6+9+12+18+27+36+54+108}{12}$  Average = $\frac{280}{12}$ Average = 23.333  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. 

Solution

The factors’ list of 108 is given as:  Factors of 108: 1, 2, 3 , 4, 6, 9, 12, 18, 27, 36, 54, and 108 Such that,  From the aforementioned list, we can clearly say that the numbers 5, 7, 11, and 13 are the four odd numbers from 1 to 15 that are not factors of 108. Images/mathematical drawings are created with GeoGebra. 

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