Factors of 9 by Prime Factorization & Factor Tree Method with Steps (2024)

In mathematics, factors are numbers that can be multiplied together to get a specific product. For example, the factors of \(12\) are \(1, 2, 3, 4, 6\), and \(12\) because they can be multiplied together in different ways to get the product \(12\) \((1 \times 12, 2 \times 6, 3 \times 4)\).

The factors of \(\textbf{9}\) are the numbers that can be multiplied together to get the product \(9\). In other words, the factors of \(9\) are the numbers that divide \(9\) without leaving a remainder. The factors of \(9\) are\(1, 3\), and \(9\).

What are Factors of 9?

Factors of a number in maths are integers that divide the number. The factors of \(\textbf{9}\) are the numbers that can be multiplied together to get the product \(9\). In other words, the factors of \(9\) are the numbers that divide \(9\) without leaving a remainder. The factors of \(9\) are \(1, 3\), and \(9\). These are the only three positive integers that can be multiplied together in different ways to get the product \(9\).

To check why \(1, 3\), and \(9\) are the factors of \(9\), we can perform a simple division. When we divide \(9\) by \(1\), we get \(9\) with no remainder. When we divide \(9\) by \(3\), we also get \(3\) with no remainder. And when we divide \(9\) by \(9\), we get \(1\) with no remainder.

Therefore, \(1\), \(3\), and \(9\) are the only factors of \(9\).

Prime Factors of 9

\(9\) is not a prime number, but it can be expressed as the product of prime numbers. The prime factorization of \(9\) is \(3 \times 3\), where \(3\) is a prime number.

To find the prime factors of \(9\), we can divide \(9\) by the smallest prime number, which is \(2\). However, \(2\) does not divide evenly into \(9\), so we move to the next smallest prime number, which is \(3\). \(3\) divides evenly into \(9\), so we can write this as \(9 = 3 \times 3\).

So, \(3\) is a prime number because it is only divisible by \(1\) and themselves. Therefore, the prime factor of \(9\) is \(3\).

Composite Factors of 9

Composite numbers can be defined as numbers that have more than two factors. Numbers that are not prime are composite numbers because they are divisible by more than two numbers.

We know that the factors of \(9\) are \(1, 3\), and \(9\). Composite numbers in mathematics are the numbers that have more than two factors. Composite factors of \(9\) is \(9\). A number can be classified as prime or composite depending on their divisibility. The number \(9\) is divisible by \(3\). So, we can say that \(9\) is a composite number and will surely have more than two factors. The composite factor of \(9\) is \(9\) only.

Pair Factors of 9

Pair factors of a number are the pairs of two numbers that when multiplied together give the original number. So, the pair factors of \(9\) are \(1\) and \(9\), \(3\) and \(3\).

When we multiply \(1\) and \(9\), we get \(9\). When we multiply \(3\) and \(3\), we also get \(9\).

• \(1 \times 9 = 9\).
• \(3 \times 3 = 3\).

Therefore, \((1, 9)\), and \((3, 3)\) are the only pairs of factors of \(9\).

Common Factors of 9

The common factors of \(9\) are the factors that \(9\) shares with another number. Since every integer has \(1\) and itself as factors, \(9\) shares the factors \(1\) and \(9\) with all other numbers.

For example, let's consider the common factors of \(9\) and \(12\). The factors of \(9\) are \(1, 3\), and \(9\), while the factors of \(12\) are \(1, 2, 3, 4, 6\), and \(12\). The common factors of \(9\) and \(12\) are the factors that both numbers share, which are \(1\) and \(3\).

Steps to Find Factors of 9

Let us understand how to find the factors of \(9\) using the below steps:

Step 1: Start by dividing \(9\) by the smallest positive integer greater than \(1\).

Step 2: Check each integer to see if it divides evenly into \(9\), this means that it is a factor of \(9\).

Step 3: The only positive integers that divide evenly into \(9\) are \(1, 3\), and \(9\). Therefore, these are the factors of \(9\).

How to Find Factors of 9?

We can find the factors of \(9\) by using below methods:

• Prime factorization of \(9\).
• Factor tree method to find factors of \(9\).
• Division method to find factors of \(9\).

Factors of 9 by Prime Factorization

Prime factorization is the process of expressing a composite number as a product of its prime factors. A composite number is any positive integer greater than \(1\) that is not a prime number.

To find the prime factorization of \(9\), we need to find the prime factors of \(9\) and express it as a product of those prime factors. Follow the steps:

Step 1: To find the prime factors of \(9\), we start by dividing it by the smallest prime number, which is \(2\). However, \(2\) does not divide evenly into \(9\).

Step 2: Next, we try dividing \(9\) by the next prime number, which is \(3\). Since \(3\) divides evenly into \(9\), we can write: \(\frac{9}{3}=3\).

Step 3: Again, divide \(3\) by the smallest prime factor, i.e., \(3\) as \(\frac{3}{3}=1\).

Step 4: Now, we cannot divide \(1\) by any prime factor.

Therefore, the prime factorization of \(9 = 3 \times 3 = 3^{2}\).

Factor Tree Method to Find Factors of 9

The factor tree method can be a useful way to visually see the prime factors of a number and to find all of the factors. The method can also be extended to larger numbers by continuing to factor each factor until only prime numbers are left.

Here are the steps to use the factor tree method to find the factors of \(9\):

Step 1: To find the factors of \(9\) using the factor tree method, we can start by writing \(9\) at the top of the tree.

Step 2: Next, we can try to find two numbers that multiply together to give \(9\). In this case, the only such pair is \(3\) and \(3\).

Step 3: Now, we have two prime factors of \(9\), and we can't break them down any further. The factors of \(9\) are simply the two prime factors, which are \(3\) and \(3\).

Step 4: Therefore, the factors of \(9\) are \(1\), \(3\), and \(9\).

Let us see how a factor tree of \(9\) looks like:

Division Method to Find Factors of 9

The division method is a systematic way of finding all the factors of a number. It is useful when dealing with larger numbers that may have many factors, but can be time-consuming for smaller numbers like \(9\), where there are only a few factors.

To use the division method to find the factors of \(9\), follow these steps:

Step 1: When we divide \(9\) by \(1, 3\), and \(9\), the remainder will be \(0\).

Step 2: At the same time, when we divide \(9\) by numbers like \(2\) or \(4\), it leaves a remainder.

Step 3: Try to divide \(9\) by the above numbers and see the results.

Therefore, the factors of \(9\) are \(1, 3\), and \(9\) by division method.

Factors of 9 Summary

• The factors of \(9\) are the numbers that can be multiplied together to get the product \(9\).
• The factors of \(9\) are \(1, 3\), and \(9\).
• \(3\) is the only prime factor of \(9\).
• The negative factors of \(9\) are \(-1, -3\), and \(-9\).
• The positive factor pairs of \(9\) are \((1, 9)\), and \((3, 3)\).
• The negative factor pairs of \(9\) are \((-1, -9)\), and \((-3, -3)\).
• The prime factorization of \(9\) is \(9 = 3 \times 3 = 3^{2}\).
• Sum of factors of \(9\) is \(13\), i.e. \(1 + 3 + 9 = 13\).

Factors of 9 Solved Examples

Example 1: Find the common factors of \(9\) and \(27\).

Solution:The factors of \(9\) are \(1, 3\), and \(9\), while the factors of \(27\) are \(1, 3, 9\), and \(27\).

So, the common factors of \(9\) and \(27\) are \(1, 3\), and \(9\).

Example2: What is the sum of all the factors of \(9\)?

Solution:The factors of \(9\) are \(1, 3\) and \(9\).

Sum \(= 1 + 3 + 9 = 13\)

Therefore, \(13\) is the required sum.

Example3: If there are \(9\) mangoes to be distributed among \(3\) children. How many mangoes does each child get?

Solution:Given,

Number of mangoes = \(9\)

Number of children = \(3\)

Each child will get = \(\frac{9}{3} = 3\) mangoes

Therefore, each children will get \(3\) mangoes each.

We hope that the above article is helpful for your understanding and exam preparations. Stay

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