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Factors of 67: Prime Factorization, Methods, Tree, and Examples

Factors of 67 are the listing of the factors that can divide the original number completely leaving no remainder behind. The factors are also called divisors. They are the numbers that divide the given number giving zero as a remainder.

It can also be defined as the product of two whole numbers such that the two whole numbers are the factor of that product and the whole numbers are exactly divisible by the product. For instance, 1 x 67 = 67.

Therefore, 1 and 67 are the factors of 67.

There can be negative as well as positive factors. Negative factors are simply the factors with the negative sign.

Another important fact about the number 67 that helps you to find the factors easily is that 67 is a prime number which means that it has only two factors 1 and 67. This is supported by the fact that prime numbers are only divisible by one and the number itself.

Some more interesting facts about the factors of 67 are in store for you. In this article, we will discover various methods for finding the factors of the number 67 along with some solved examples.

What are the factors of 67?

The factors of 67 are 1 and 67, as they can both divide 67 without leaving any residue or remainder. The factors of 67 are those numbers that are exactly divisible by 67, leaving the remainder zero.

In simple words, it can also be explained as the pair of numbers that, when multiplied, give 67 as their product.

How To Calculate the Factors of 67?

To calculate the Factors of 67, first consider the number whose factors are to be found (67) and start dividing the number 67 by various natural numbers such as:

\[ \dfrac {67}{1} = 67 \]

\[ \dfrac{67}{2} = 33.5 \]

\[ \dfrac{67}{11} = 6.09 \]

\[ \dfrac{67}{67} = 1 \]

After all the trials, the only likelihood of divisibility of 67 is by two integers that are 1 and 67.

You can also find the factors by considering the fact that 67 is a prime number and that indicates that it can only have two factors i.e. 1 and 67. Another method of determining the factors of 67 is to figure out two numbers whose product will be equal to 67, and those two numbers are 1 and 67.

Therefore, by checking all the possibilities, it can be concluded that the only factors of 67 are 1 and 67 itself.

The factors of 67 are 1 and 67.

Here are some fun facts regarding the factors of the number 67:

  1. 67 is a prime number, therefore it is only divisible by 1 and 67 itself.
  2. The number 67 does not come in any table except for the table of 1 and 67.
  3. 67 is also an odd number, thus it can not be divided by two.
  4. A fun fact about the number 67 is that it is a special number. It can be obtained by adding five consecutive prime numbers which are 7, 11, 13, 17, and 19.
  5. The sum of the factors of 67 is an even number that is 68.

Let’s explore some important properties of the factors of 67.

In this section are some of the important points that must be considered while finding the factors of a given number:

  1. The factors of a number can never be in the form of decimals or fractions, so the factors of 67 are neither in the decimal form nor a fraction.
  2. If the whole number is a factor of a certain number, then its additive inverse is also its factor called a negative factor. For instance, if 67 is the factor of 67, then -67 is also the factor of the number 67.
  3. 1 is the factor of every whole number, therefore 1 is also the factor of 67.
  4. Multiplication and division techniques are used to find the factors of the whole numbers.

Factors of 67 by Prime Factorization

Prime Factorization is the technique of finding the prime factors by configuring which pair of prime factors will be multiplied together such that their product gives a certain number whose factors are to be determined.  

The prime factorization process goes on until the number 1 is obtained as a quotient. Also, remember in prime factorization every factor is a prime number.

The prime factorization of 67 is given as:

Figure 1

Factor Tree of 67

The factor tree of the number 67 is given as follows:

Figure 2

The factor tree is the representation of the factor of a number specifically obtained by prime factorization. Each branch of the tree extends to produce the factors until no more factorization is possible. The end of the branch is always a prime number.

For the number 67, there are only two factors: 1 and 67, as 67 is also a prime number itself.

Factors of 67 in Pairs

The pair factors are the pair of factors that gives the actual number when multiplied together. 

As mentioned above, 67 is a prime number, therefore it has only two factors, so the possible pair factors of 67 are given as follows:

67 x 1 = 67 

So, the pair of factors is ( 1, 67).

 1 x 67 = 67 

The other possible pair of factors is ( 67, 1 ).

The same explanation stands for negative pair of factors as well. So the negative pair of factors can be obtained as:

– 67 x – 1 = 67 

– 1 x – 67 = 67 

Therefore, the possible negative pair of factors are (-1, -67) or (-67, -1).

Factors of 67 Solved Examples

Let’s solve some examples to better understand the concepts and techniques used for finding the factors of 67 and how the factors are used to solve various mathematical problems.

Example 1

Find the common factors of 67 and 201. Solve using the technique of prime factorization.

Solution

First, find the prime factor of 67 using prime factorization.

The prime factors of 67 are 1 and 67.

Now, find the factors of 201 using prime factorization.

The factors of 201 are 1, 3, 67, and 201.

So, the common factors of 67 and 201 are given 1 and 67.

Example 2

Calculate the average of the factors of the number 67.

Solution

The factors of 67 are 1 and 67 as 67 is a prime number.

The formula for calculating the average of the number is given:

\[ Average =  \dfrac{\text{Sum of the number}}{\text{Total number of the values }} \]

The total number of factors of 67 is two.

Therefore, the average value of factors of 67 is calculated as:

\[ Average = \dfrac{ 1 + 67 }{2} \]

\[ Average = \dfrac{ 68 }{ 2 } \]

Average = 34 

Therefore, the average of all the factors of 67 is 34.

Example 3

Fallon needs to find a factor of 67 that is greater than 60. Help her find the number.

Solution

The factors of 67 are 1 and 67.

The factor greater than 60 is 67.

Hence, the number Fallon looking for is 67.

 Images/mathematical drawings are created with GeoGebra. 

Factors of 66 | Factors List | Factors of 68

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