## Define the Factors of 9

Factors of 9, what it really defines is that all of the positive and negative integers (whole numbers) that can be evenly divided into 9. The integer 9 has 3 factors i.e. 1, 3, and 9. If 9 is divided by any of the three factors then the answer will be a whole number.

For Example, $9 \div 3 =3$.

## Factors of 9 Definition

Factors of any number are those natural numbers that can divide the original number, without leaving any remainder. A factor of 9 is a number that divides the number 9 without leaving any remainder. The factor of 9 can be referred to as numbers or algebraic expressions that evenly divide the number 9. The factors of a number 9 can either be positive or negative.

For example, the number 9 can be factored in as 1, 3 and 9. Also, we know the product of two negative numbers is a positive number only. Therefore, the factors of number 9 are 1, -1, 3, -3, 9, and -9.

## How To Find the Factors of 9?

The factor of 9 can be calculated either by the division method or the prime factorization method. Let us first discuss how to find the factors of 9 using the division method.

To find the factor of 9 using the division method, we will divide the number 9 by the natural number and check which numbers can exactly divide 9 and leave the remainder zero.

Step 1: When the number 9 is divided by 1, 3, and 9, the remainder will be 0.

Step 2: At the same time, when 9 is divided by other numbers like 2 or 4, it obtains the remainder.

Step 3: Now, try to divide 9 by the numbers and check the results.

$9 \div 1$ = Quotients = 9, Remainder = 0

$9 \div 2$ = Quotients = 4, Remainder = 1

$9 \div 3$ = Quotients = 3, Remainder = 0

$9 \div 4$ = Quotients = 2, Remainder = 1

$9 \div 5$ = Quotients = 1, Remainder = 4

$9 \div 6$ = Quotients = 1, Remainder = 3

$9 \div 7$ = Quotients = 1, Remainder = 2

$9 \div 8$ = Quotients = 1, Remainder = 1

$9 \div 9$ = Quotients = 1, Remainder = 0

Therefore, the numbers 1, 3, and 9 are the factors of 9.

## How To Find Factor of 9 Using the Prime Factorization Method?

In the prime factorization method, we will divide the number 9 by prime numbers and see if it does not leave any remainder. We then continue division with the quotient. In this way, the number 9 can be expressed as a product of its prime factors. We can find the prime factors of 9 by both the division method and factor tree.

## Prime Factorization of 9 Using Division Method

To find the prime factors of 9 using the division method, we need to divide by each of the prime numbers till we get the remainder as 1.

Step 1: Divide the number 9 by the smallest prime factor. As we know, 2 is the smallest prime number but if we divide 9 by 2 we will get a fraction. Therefore, we will now consider the next prime number i.e. 3

For Example:

$9 \div 3 =3$

Step 2: Now, we will divide the number 3 by itself which gives us the quotient as 1.

$3 \div 3 =1$

Step 3: As the number 1 cannot be divided by any prime factor. Therefore,

Prime Factorization of 9 is $9 = 3 \times 3$.

## Prime Factor Tree For 9

A prime factor tree for 9 provides the pictorial representation of the prime factorization of the positive integer 9. Starting with the given integer i.e. 9 at the top of the tree, two branches are drawn towards the positive factors of 9. This process is repeated for the numbers located at the end of the branch that is drawn until each leaf of the factor tree for 9 is a prime number.

The factor tree for 9 is shown below:

Multiplying all the leaves of the tree gives $9 = 3 \times 3$.

## Factor Pairs of 9

A factor pair of 9 is a combination of two factors that can be multiplied together to get the number equals 9. For the number 9, all of the positive factors are listed below:

$ 1 \times 9 =9$

$ 3 \times 3 =9$

We can also list out the negative factors pairs of 9 as shown below:

$ -1 \times -9 =9$

$ -3 \times -3 =9$

Notice in the negative factor pairs of 9, we are multiplying the minus with minus. Therefore, the resultant number is positive.

## Solved Examples

1. What is the product of all the factors of 9?

Solution:

As we know, the factors of 9 are 1, 3, and 9. Therefore, its product will be $1 \times 3\times 9 =27$.

2. Find the greatest common factors of 7 and 9.

Solution:

Factors of 7 are 1, and 7

Factors of 9 are 1, 3, and 9.

Both 7 and 9 have only one common factor i.e. 1. This implies both 7 and 9 are coprime.

## Practise Problems

**1. What is the****average**** factors of 9**

a) $\dfrac{11}{3}$

b) $\dfrac{21}{3}$

c) $\dfrac{13}{3}$

d) $\dfrac{11}{2}$

**2. Nikitha was playing a math game with her friends. In the game, he was asked to find the factors of 9. What will be his answer?**

a) [1,3,9]

b) [1.2.9]

c) [3,5,9]

d) [6,9,3]

**3. Kiran's class teacher asked him to calculate the sum of all the factors of 9. Can you help him find the sum?**

a) 13

b) 12

c) 10

d) 9

Answers

1) c

2) a

3) a

## Conclusion

Here we have learnt about the definition of factor and we found about the factors of 9 by using different methods like division method,prime factorisation,factor pair and factor tree. By using the same method and steps you can also find factors for other numbers.

## FAQs on Factors of 9

1. Are both the multiples and factors of 9 the same?

Multiples of 9 are the numbers that are divisible by the original number whereas the factors of 9 are the numbers that divide the original number evenly. Therefore, both are not the same.

2. What would be the sum of all the factors of 9?

As we know, the factors of 9 are 1,3, and 9. Therefore, its sum will be 1 + 3 + 9 = 13.

3. How many factors of 9 are also the factors of 27?

As we know, the factors of 9 are 1, 3, and 9 and the factors of 27 are 1, 3, 9, and 27. Therefore, [1,3, and 9] are the common factors of 27.