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Factors of 231: Prime Factorization, Methods, and Examples

The factors of 231 are numbers that give off zero as the remainder when 231 acts as the divided. Since the number 231 is an odd composite number so it consists of multiple factors. 

These factors can be both positive and negative and are determined through various methods. 

Factors of 231

Here are the factors of number 231.

Factors of 231: 1, 3, 7, 11, 21, 33, 77, 231

Negative Factors of 231

The negative factors of 231 are similar to its positive factors, just with a negative sign.

Negative Factors of 231: -1, -3, -7, -11, -21, -33, -77, and -231

Prime Factorization of 231

The prime factorization of 231 is the way of expressing its prime factors in the product form.

Prime Factorization: 3 x 7 x 11

In this article, we will learn about the factors of 231 and how to find them using various techniques such as upside-down division, prime factorization, and factor tree.

What Are the Factors of 231?

The factors of 231 are 1, 3, 7, 11, 21, 33, 77, and 231. All of these numbers are the factors as they do not leave any remainder when divided by 231.

The factors of 231 are classified as prime numbers and composite numbers. The prime factors of the number 231 can be determined using the technique of prime factorization.

How To Find the Factors of 231?

You can find the factors of 231 by using the rules of divisibility. The divisibility rule states that any number, when divided by any other natural number, is said to be divisible by the number if the quotient is the whole number and the resulting remainder is zero.

To find the factors of 231, create a list containing the numbers that are exactly divisible by 231 with zero remainders. One important thing to note is that 1 and 231 are the 231’s factors as every natural number has 1 and the number itself as its factor.

1 is also called the universal factor of every number. The factors of 231 are determined as follows:

\[\dfrac{231}{1} = 231\]

\[\dfrac{231}{3} = 77\]

\[\dfrac{231}{7} = 33\]

\[\dfrac{231}{11} = 21\]

\[\dfrac{231}{21} = 11\]

\[\dfrac{231}{33} = 7\]

\[\dfrac{231}{77} = 3\]

\[\dfrac{231}{231}=1\]

Therefore, 1, 3, 7, 11, 21, 33, 77, and 231 are the factors of 231.

Total Number of Factors of 231

For 231 there are 8 positive factors and 8 negative ones. So in total, there are 16 factors of 231. 

To find the total number of factors of the given number, follow the procedure mentioned below:

  1. Find the factorization of the given number.
  2. Demonstrate the prime factorization of the number in the form of exponent form.
  3. Add 1 to each of the exponents of the prime factor.
  4. Now, multiply the resulting exponents together. This obtained product is equivalent to the total number of factors of the given number.

By following this procedure the total number of factors of 231 is given as:

Factorization of 231 is 1 x 3 x 7 x 11

The exponent of all factors is 1.

Adding 1 to each and multiplying them together results in 16.

Therefore, the total number of factors of 231 is 16. 8 are positive and 8 factors are negative.

Important Notes

Here are some important points that must be considered while finding the factors of any given number:

  • The factor of any given number must be a whole number.
  • The factors of the number cannot be in the form of decimals or fractions.
  • Factors can be positive as well as negative.
  • Negative factors are the additive inverse of the positive factors of a given number.
  • The factor of a number cannot be greater than that number.
  • Every even number has 2 as its prime factor, the smallest prime factor.

Factors of 231 by Prime Factorization

The number 231 is a composite number. Prime factorization is a useful technique for finding the number’s prime factors and expressing the number as the product of its prime factors.

Before finding the factors of 231 using prime factorization, let us find out what prime factors are. Prime factors are the factors of any given number that are only divisible by 1 and themselves.

To start the prime factorization of 231, start dividing by its smallest prime factor. First, determine that the given number is either even or odd. If it is an even number, then 2 will be the smallest prime factor.

Continue splitting the quotient obtained until 1 is received as the quotient. The prime factorization of 231 can be expressed as:

231 = 3 x 7 x 11

Factors of 231 in Pairs

The factor pairs are the duplet of numbers that when multiplied together result in the factorized number. Factor pairs can be more than one depending on the total number of factors of the given numbers.

For 231, the factor pairs can be found as:

1  x 231 = 231

 3 x 77 = 231

7 x 33 = 231

11 x 21 = 231

The possible factor pairs of 231 are given as (1, 231), (3, 77), (7, 33), and (11, 21).

All these numbers in pairs, when multiplied, give 231 as the product.

The negative factor pairs of 231 are given as:

-1  x -231 = 231

 -3 x -77 = 231

-7 x -33 = 231

-11 x -21 = 231

It is important to note that in negative factor pairs, the minus sign has been multiplied by the minus sign due to which the resulting product is the original positive number. Therefore, -1, -3, -7, -11, -21, -33, -77, and -231 are called negative factors of 231.

The list of all the factors of 231 including positive as well as negative numbers is given below.

Factor list of 231: 1, -1, 3, -3, 7, -7, 11, -11, 21, -21, 33, -33, 77, -77, 231, and -231

Factors of 231 Solved Examples

To better understand the concept of factors, let’s solve some examples.

Example 1

How many factors of 231 are there?

Solution

The total number of Factors of 231 is 8.

Factors of 231 are 1, 3, 7, 11, 21, 33, 77, and 231.

Example 2

Find the factors of 231 using prime factorization.

Solution

The prime factorization of 231 is given as:

231 $\div$ 3 = 77 

77 $\div$ 11 = 7 

7 $\div$ 7 =1

So the prime factorization of 231 can be written as:

3 x 7 x 11 = 231 

Factors of 230|Factors List| Factors of 232

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