Factors of 108

Factors of 108 are the numbers that when multiplied in pairs give the product as 108.

The factors of 108 consist of all the integers i.e. positive as well as negative whole numbers – 108 can be divided into such integers.

Before we study about factors of 108, let’s have a quick recall about the term “Factor”.

What is a Factor?

A factor is a Latin word, and it means “a doer” or “a performer” or “a maker”.

A number is said to be a factor when the first number divides the second number completely without leaving any remainder.

If put into other words, then multiplying 2 whole numbers provides us a product, then the numbers we are multiplying with each other are said to be the factors of the product.

This is simply because they are divisible by the product. Every number has minimal 2 factors – the number itself and a common factor – that is 1 (one). The number of factors depends on the type of number.

Two Types Of Numbers –

A composite no is a no that possesses over 2 factors.

On the other hand, a prime number is a number that consists of only two factors.

Methods to find factor

Multiplication and division are two methods of finding factors. In addition, we can also apply the divisibility rules.

Factors by Division

By using the division method, steps to find the factors of a number are –

  • You are required to determine all the numbers equal to or less than the given number.
  • Divide the given number by each of the provided numbers.
  • The divisors that provide the remainder as 0 are the factors of the given number.

Factors by Multiplication

Steps to find the factors by using the multiplication method are –

  • Write the number (whose factors are to be determined) as the product of 2 numbers in different possible ways.
  • All the nos that are involved in all these products are known as the factors of the given number.

Properties of factors

  • Factors are found by employing division and multiplication are the operations that are used
  • Except for 0 and 1, every integer has a minimum of two factors i.e. one and the number itself.
  • All integers have a definite no of factors.
  • The factor of a no can be only equal or less than the number itself. It can never be greater than the number whose factors are to be determined.

Fun facts about factors

  • Factors are only whole numbers or integers, they can never be fractions or decimals.
  • Factoring is a typical method to solve or simplify algebraic expressions.
  • All even no always have two as its factor.
  • All no greater than zero and ending in a zero are ten, five, and two.
  • 5 will always be a factor for all numbers that end in 5 and 0.

Prime factorization

The prime factorization is an outcome of factoring a number into a set of constituents in which every number is a prime number.

Each and every number in prime factorization is a prime number.  

To write the number as a product of prime factors, we might often have to repeat the factors too.

Example: To write the prime factorization of 16, we can write 16 = 2 ✕ 2 ✕ 2 ✕ 2, i.e, the prime factor 2  is repeated four times.

Real-life applications of factorization

Understanding the factorization process allows you to easily navigate number relationships in real-life without relying on your phone or calculator to do the work for you.

  • Equal division

Tip – When you have to divide something into equal pieces, look at no other option than the factorization process.

For example, if six people come together to eat a whole pizza that has been cut into 30 slices, it would only be fair if everyone received the same number of slices.

Therefore, this pizza can be divided into equal shares because 6 (the number of people) is a factor of 30(the number of pizza slices). Dividing 30 by 6 gives a result of 5, so each person gets 5 slices.

  • Traveling with Factors

Factors are also useful when traveling. If you need to travel 600 miles on a vacation, then you are required to know how many hours you must drive so you can plan your trip.

At an average speed of 60 mph, it would take 10 hours to get to your destination (60 x 10 = 600).

  • Factoring with money

Exchanging money and its divisions into smaller units relies heavily on factoring.

You probably already know that four quarters equal one dollar in America. In India, a rupee was further divided into 1 paisa, 5 paisa, 10 paisa, 25 paisa, and 50 paisa.

If we talk about the number 108, factors of this number are the integers that can divide the given number evenly without leaving any remainder.

What are the Factors of 108?

The number 108 is an even composite number has many factors other than 1 and itself. They are 3, 4, 6, 9, 12, 18, 27, 36, 54   Hence, there are a total of twelve factors of 108.

Pair Factors of 108

The numbers that are multiplied in pairs resulting in an original number are the pair factors of 108.

The pair factors of 108 can be represented in positive as well as negative factors.

Now we are aware of all the factors of the number in consideration i.e. 108, we can list down all such factors and write down all the different combinations that can be multiplied with each other to get 108.

For instance, the factor pair of 108 can be (108, 1) and (-108, -1). For negative pair factors, minus times minus equals plus.

Hence, it results in positive pair factors. Thus, the complete list of positive and negative pair factors of 108 is given below:-

Positive Pair Factors of 108

(9, 12); (6, 18); (4, 27); (3, 36); (2, 54); (1; 108)

Negative Pair Factors of 108

(-9, -12); (-6, -18); (-4, -27); (-3, -36); (-2, -54); (-1, -108)

How to Find Factors of 108?

We can evaluate the factors of 108 by the division method.

In this method, the factors of 108 are found by dividing 108 by different consecutive integers.

If the integers divide 108 completely and leave the remainder 0, then the integers are the factors of 108.

The table mentioned below determines the factors of 108 by employing the division method –

108/1 = 108 (Factor is One and Remainder is Zero)

108/2 = 54 (Factor is Two here and Remainder is Zero )

108/3 = 36 (Factor Is Three and Remainder Is Zero)

108/4 = 27 (Factor Is Four and Remainder Is Zero)

108/6 = 18 (Factor is Six and Remainder Is Again Zero Since It’s A Factor)

108/9 = 12 (Factor = 9 and Remainder = 0)

108/12 = 9 (Factor Is Twelve and Remainder Is Zero)

108/18 = 6 (Factor = 18, Remainder = 0)

108/27 = 4 (Factor = 27, Remainder = 0)

108/36 = 3 (Factor Is Thirty-Six and Remainder is Zero)

108/54 = 2 (Factor is Fifty-Four and Remainder Is Zero)

108/108 = 1 (Factor Is One Hundred And Eight and Remainder Is Zero)

Note: If we divide 108 by any numbers other than 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108, then it leaves a remainder.

Hence, the factors 108 are

  • 1 (One)
  • 2 (Two)
  • 3 (Three)
  • 4 (Four)
  • 6 (Six)
  • 9  (Nine)
  • 12 (Twelve)
  • 18 (Eighteen)
  • 27 (Twenty-Seven)
  • 36 (Thirty-Six)
  • 54 (Fifty Four), and
  • 108 (One Hundred And Eight i.e. The Number Itself).

Prime Factors Of 108

As we denote 108 as a product of its prime factor, then the smallest prime number in the series is known as the smallest prime factor.

Hereinafter laid down are the steps to find the prime factors –

  • The first step is to divide the number 108 with the smallest prime factor, say 2. Mathematically, 108 ÷ 2 = 54
  • Then, divide the result i.e. 54 again by 2. Mathematically, 54 ÷ 2 = 27
  • Now if we will divide 27 by 2, we will get a fractional number.A factor cannot be a decimal or a fractional number. So, move forward with the next prime factor, i.e., 3
    •  27 ÷ 3 = 9
  • Divide again by 3. 
    • 9 ÷ 3 = 3
  • And againà3 ÷ 3 = 1
  • Thus, we receive the number 1 at the end of the division process. This is the terminating condition, and we cannot proceed further.

Hence, the prime factors of 108 are written as  2 × 2 × 3 × 3 × 3, where 2 and 3 are prime numbers.

Mathematically,

108 = 2² x 3³.

Factor Tree Of 108

The factor tree of 108 is another method for producing all the factors (including prime factorization) of a given number.

How To Carry Out The Factor Tree Of A Given Number

  • You are required to write the number at the top of your solution paper and it will be the tip of your tree.
  • Deduce the tree for its factors by drawing two downward diagonal lines beneath the number. One should point right and the other should point left.
  • Pick any pair of factors for the no you tend to work with. The product of the 2nos should be equal to your original no when multiplied together in order to qualify as a pair of factors. These factors will make the first branch of your factor tree. Pick any two factors. The end result will always be the same, no matter which one you start with.

Note:-If there are 0 factors that equal the original number when multiplied together (other than the number itself and the number ‘1’), the number cannot be converted into the factor tree and is a prime number.

  • You are required to break down every set into its factors. Break down your 1st 2 factors into their own sets of 2 factors apiece.
  • As before, 2 numbers can only be considered factors if they equal the current value when multiplied together.
  • Don’t break down prime nos any further.
  • Repeat unless you reach nothing but prime numbers. You will be required to break down each no as far as possible unless you separate it into nothing but prime nos.
  • Create as many branches as requisite in the procedure.

Note:- There has to be the number ‘1’ present anywhere in your tree.

The number 108 is a composite number because 108 can be divided by one, by itself or at least by two and three. So, it is plausible to draw its prime tree.

Frequently Asked Questions

  • Is 27 a factor of 108?

Yes, 27 is a factor of 108. As twenty-seven divides one hundred and eight exactly and leaves the remainder zero, twenty-seven is a factor of one hundred and eight.

  • Is 108 a square number?

No! One hundred and eight is NOT a square no, because the square root of one hundred and eight is (10.39) which is not an integer, but one hundred and eight is a whole no.

  • What is the amount equal to the sum of all the factors of one hundred and eight, including one hundred and eight?

Factors of 108 are given below: 

  • 1 (One)
  • 2 (Two)
  • 3 (Three)
  • 4 (Four)
  • 6 (Six)
  • 9  (Nine)
  • 12 (Twelve)
  • 18 (Eighteen)
  • 27 (Twenty-Seven)
  • 36 (Thirty Six)
  • 54 (Fifty Four), and
  • 108 (One Hundred And Eight i.e. The Number Itself).  

Now, sum up all the factors and that comes

= 280

Hence, the sum of all factors of 108 = 280

  • What is the highest or greatest common factor of one hundred and eight?

The highest common factor also referred to as the greatest common factor of two numbers can be identified by establishing a comparison of the two numbers and taking the highest common prime factor, the prime factorization. The GCF is part of the common prime factors of two numbers and it is 1 if there is no common factor.