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Factors of 189: Prime Factorization, Methods, and Examples

The factors of 189 are the whole number that wholly divides the number. The number 189 is odd and composite. It means that it has more than two factors

When any two integer factors are multiplied, it results in the actual number irrespective of their signs.

Factors of 189

Here are the factors of number 189.

Factors of 189: 1, 3, 7, 9, 21, 27, 63, and 189

Negative Factors of 189

The negative factors of 189 are similar to its positive factors, just with a negative sign.

Negative Factors of 189: -1, -3, -7, -9, -21, -27, -63 and -189

Prime Factorization of 189

The prime factorization of 189 is the way of expressing its prime factors in the product form.

Prime Factorization: 3 x 3 x 3 x 7

In this article, we will learn about the factors of 189 and how to find them using various techniques such as upside-down division, prime factorization, and factor tree.

What Are the Factors of 189?

The factors of 189 are 1, 3, 7, 9, 21, 27, 29, and 189. All of these numbers are the factors as they do not leave any remainder when divided by 189.

The factors of 189 are classified as prime numbers and composite numbers. The prime factors of the number 189 can be determined using the technique of prime factorization.

How To Find the Factors of 189?

You can find the factors of 189 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 189, create a list containing the numbers that are exactly divisible by 189 with zero remainders. One important thing to note is that 1 and 189 are the 189’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 189 are determined as follows:

\[\dfrac{189}{1} = X\]

\[\dfrac{189}{3} = 63\]

\[\dfrac{189}{7} = 27\]

\[\dfrac{189}{189} = 1\]

Therefore, 1, 3, 7, 9, 21, 27, 63, and 189 are the factors of 189.

Total Number of Factors of 189

For 189 there are 8 positive factors and 8 negative ones. So in total, there are 16 factors of 189. 

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 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 X is given as:

Factorization of X is 1 x 3^{3} x 7.

The exponent of 1, and 7 is 1 while factor 3 has an exponent of 3.

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

Therefore, the total number of factors of 189 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 189 by Prime Factorization

The number 189 is a prime 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 189 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 189, 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 189 can be expressed as:

\[  X = 3 \times 3 \times 3 \times  7\]

Factors of 189 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 189, the factor pairs can be found as:

\[ 1 \times 189 = 189 \]

\[ 3 \times 63 = 189 \]

\[ 7 \times 27 = 189 \]

\[ 9 \times 21 = 189 \]

The possible factor pairs of X are given as (1, 189), (3, 63), (7, 27), and (9, 21).

All these numbers in pairs, when multiplied, give 189 as the product.

The negative factor pairs of 189 are given as:

\[ -1 \times -189 = 189 \]

\[ -3 \times -63 = 189 \]

\[ -7 \times -27 = 189 \]

\[ -9 \times -21 = 189 \]

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, -9, -21, -27, -63 and -189 are called negative factors of 189.

The list of all the factors of 189 including positive as well as negative numbers is given below.

Factor list of X: 1, -1, 3, -3, 7, -7, 9, -9, 21, -21, 27, -27, 63, -63, 189, and -189

Factors of 189 Solved Examples

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

Example 1

How many factors of 189 are there?

Solution

The total number of Factors of 189 is 8.

Factors of 189 are 1, 3, 7, 9, 21,27, 63 and 189.

Example 2

Find the factors of 189 using prime factorization.

Solution

The prime factorization of X is given as:

\[ 189 \div 3 = 63 \]

\[ 63 \div 3 = 21 \]

\[ 21 \div 3 = 7 \]

\[ 7 \div 7 = 1 \]

So the prime factorization of 189 can be written as:

\[ 3^{3} \times 7 = 189 \]

Factors of 188|Factors List| Factors of 190

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