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Factors of 60: Prime Factorization, Methods, Tree, and Examples

Factors of 60 are the numbers that evenly divide 60, leaving the remainders as zero. Factors of a number can be positive as well as negative. Positive and negative factors are the same but have opposite signs. 

The easiest method to find factors is the multiplication method. Find two numbers whose product is equal to 60. Both of the numbers will be the factors of 60.

In this article, we will cover every side of the factors of 60, the different techniques to discover them, how to fabricate a factor tree, and some properties of factors. In addition, there are some solved examples for better understanding. 

What Are the Factors of 60?

The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. Number 60 is evenly divisible by all these integers. 

60 has twelve positive factors. By multiplying these integers in such pairs that their result must be equal to 60, those numbers are said to be the factor pairs of 60.

How To Calculate the Factors of 60?

You can calculate the factors of 60 by using the division method. The rule we must follow is that the remainder of the division must be zero. 

There are two most common methods to find the factors of a number.

  1. Division Method.
  2. Multiplication Method.       

The division method is discussed below:

60 is a composite number because it has more than 2 factors. As we know that numbers on the number line between 1 to 60 and -1 to -60 that evenly divide 60 will be the factors of 60. Start dividing it by different numbers and check for each positive and negative number between 1 and 60. The number will be the factor of 60 only if the remainder of the division is zero.

Starting with number one. Number 1 is a factor of every number because every number is divisible by 1, leaving the remainder zero. 

\[\frac {60}{1}= 60\]

1 and -1, both are factors of 60. 

60 is an even composite number, so it can be evenly divided by 2. 

\[\frac {60}{2}= 30\]

2, -2, 30, and -30 are also factors of 60.

Dividing 60 by 3 gives: 

\[\frac {60}{3}= 20\]

The remainder is 0.

3, -3, 20, and -20 are also factors of 60.

Now divide 60 by 4:

\[\frac {60}{4}= 15\]

The remainder is zero, so 4, -4, 15, and -15 are also the factors of 60.

Checking for 5:

\[\frac {60}{5}= 12\]

5, -5, 12, and -12 are also the factors of 60.

Dividing 60 by 6 gives:

\[\frac {60}{6}= 10\]

6, -6, 10, and -10 are also the factors of 60.

Every number divides itself evenly, leaving the remainder zero. It means every number is a factor and a multiple in itself. 

By the above calculations, we culminate the factors list of 60 as given below:

Positive  factors of 60 = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 

Negative  factors of 60 = -1, -2, -3, -4, -5, -6, -10, -12, -15, -20, -30, -60 

Properties of factors:

  • Factors are always whole numbers, and can not be written in p/q form. In other words, factors can never be in the form of fractions or decimals. 
  • Every integer has a unique prime factorization expression. 
  • All even numbers have 2 as their factor.
  • Every number contains a finite number of factors.
  • A factor of a number can never be greater than the number itself.
  • A number having more than two factors is known as a composite number.
  • If a number has only two factors, the number is a prime number.

Factors of 60 by Prime Factorization

Prime factorization means breaking a composite number into prime numbers that are its factors. By multiplying these prime numbers, if the product is equal to 60, the multiplicands are known as prime factors of 60.

The two common ways to find prime factorization are:

  1. Factor Tree.
  2. Division Method.

We are going to discuss the division method. Start dividing 60 by the smallest prime factor, 1 is not a prime number. 2 will be considered as the smallest prime factor. 

\[\frac {60}{2}= 30\]

Divide it by 2 because it is further divisible.

\[\frac {30}{2}= 15\]

15 is not divisible by 2. Now divide it by the next prime number, which is 3.

\[\frac {15}{3}= 5\]

Again, divide with the next prime factor because 5 is not divisible by 3. The next prime factor is 5.

\[\frac {5}{5}= 1\]

Prime factorization of 60 is shown below in figure 1:

Figure 1

The prime factorization of 60 is shown below:

\[ 2 \times 2 \times 3 \times 5 = 60 \]

This can also be written as 

\[ 2^2 \times 3 \times 5 = 60 \]

Factor Tree of 60

The factor tree is a special diagram that expresses a number in the form of its prime factors. A factor tree is a pictorial representation.

It consists of the number at the top; further, splits into two branches, one consisting of a quotient, and the other one consisting of a devisor. The quotient will be further divided and branched. The process of division goes on until you can’t make further factors. 

The factor tree of 60 is shown below as:

Figure 2

We are dividing the 60 into its possible factors. Divide 60 by 2 quotient will be 30, where 2 is the prime number, so it cannot be factored further. Now we will further factorize 30 and divide 30 by 2 quotient will be 15. Again, splitting 15 gives 3 and 5.

Factors of 60 in Pairs

Factor pairs are factors of the given number. We multiply those factors so that their product is equal to the original number. A set of two factors, when multiplied together, gives a particular number that is equal to the original number.

Factors when multiplied to give product 60, will be known as factor pairs of 60

\[ 3 \times 20= 60 \]

60 is the product of 3 and 20. In other words, 60 is a multiple of 3 and 20. Hence, 3 and 20 are factor pairs of 60. 

\[ 4 \times 20= 80 \]

4 and 20 both are factors of 60, but when multiplied the product is not equal to 60. Hence, they are not a factor pair of 60. 

Positive factor pairs of 60 are as follows:

\[ 1 \times 60= 60 \]

\[ 2 \times 30= 60 \]

\[ 3 \times 20= 60 \]

\[ 4 \times 15= 60 \]

\[ 5 \times 12= 60 \]

\[ 6 \times 10= 60 \]

By looking at the above multiplication, we will write the factor pairs for 60 as (1, 60), (2, 30), (3, 20), (4, 15), (5, 12), and (6, 10). 

Negative factor pairs of 60 are as follows:

\[ -1 \times -60= 60 \]

\[ -2 \times -30= 60 \]

\[ -3 \times -20= 60 \]

\[ -4 \times -15= 60 \]

\[ -5 \times -12= 60 \]

\[ -6 \times -10= 60 \]

When a negative sign is multiplied by a negative sign, the product is always positive. 

The negative factor pairs are (-1, -60), (-2, -30), (-3, -20), (-4, -15), (-5, -12), and (-6, -10),

Factors of 60 Solved Examples

For further understanding, here are some solved examples of factors of 60.

Example 1

Find the range of factors of 60.

Solution

First of all, list the factors of 60. Keep in mind factors should be in ascending order

Factors of 60 = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

The formula to calculate the range is as follows:

Range = Max Value – Min Value

The max value means the greatest number in the factors list, and the min value is the lowest number in the factor list.

Max value: 60

Min value: 1

Now putting the values in the formula of range

Range = 60-1

Range = 59 

The range for factors of 60 is 59

Example 2

Find the common factors of 40 and 60.

Solution

Firstly, list the factors of 40 and 60.

Factors of 40 are:

Factors of 40 = 1, 2, 4, 5, 8, 10, 20, 40

Factors of 60 are:

Factors of 60 = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Common factors are the factors that are present in both lists of factors. 

Common factors of 40 and 60 are:

Common factors are = 1, 2, 4, 5, 10, 20

Example 3

Jony bought 60 candies for his birthday party. The cost of one candy was 2$. Calculate the total cost of 60 candies. He made X number of goody bags, he placed 5 candies in each goody bag. Also, calculate how many goody bags he made.

Solution

Cost of one candy = 2

Total candies he bought = 60

The total cost will be:

Total cost: 2 x 60 =120

Candies in each bag= 5

Total goody bags= X

\[\frac {60}{5}= 12\]

Jony made 12 goody bags for his birthday party.

Images/mathematical drawings are created with GeoGebra.

Factors of 59|Factors List| Factors of 61

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