What are the Factors of 9? List, Factor Tree & Prime Factorization
The concept of factors of 9 is a building block in understanding number operations in Maths. Whether preparing for exams or solving real-world division problems, knowing the factors of 9 helps you approach questions with confidence and speed.
What Are the Factors of 9?
A factor of 9 is a whole number that divides 9 without leaving any remainder. In other words, if you can multiply two whole numbers to get 9, those numbers are factors. This concept appears in topics like divisibility, multiples, and number patterns. Factors help us break down numbers, solve equations, and even find HCF/LCM in advanced chapters.
Key Formula for Factors of 9
Here’s the simple formula for finding factors: If \( 9 \div n = \) whole number (no remainder), then n is a factor of 9.
List of All Factors of 9
The factors of 9 are: 1, 3, and 9.
- 1 × 9 = 9
- 3 × 3 = 9
So, 1, 3, and 9 are the only positive whole numbers that divide 9 exactly.
How to Find Factors of 9
To find the factors of 9, use these easy steps:
- Start with 1. \( 9 \div 1 = 9 \) (No remainder)
- Try 2. \( 9 \div 2 = 4.5 \) (Not whole, so not a factor)
- Try 3. \( 9 \div 3 = 3 \) (No remainder)
- Try numbers up to 9. Only 1, 3, and 9 will divide 9 exactly.
Negative numbers can also be factors, so -1, -3, and -9 are included in advanced topics.
Prime Factorization of 9
The prime factors of 9 are the prime numbers that multiply to 9. Here’s how you find them:
- Divide by the smallest prime: \( 9 \div 3 = 3 \)
- Divide by 3 again: \( 3 \div 3 = 1 \)
So, the prime factorization of 9 is \( 3 \times 3 \).
Factor Tree of 9
A factor tree helps you visualize the prime factorization:
- Start with 9 at the top.
- Break it as 3 × 3 (both are prime, so tree ends).
The tree shows: 9 → 3 × 3.
Properties of Factors of 9
| Property | 9 |
|---|---|
| Is 9 a prime number? | No, 9 is composite. |
| How many positive factors? | 3 (1, 3, 9) |
| Are all factors odd? | Yes |
| Factor pairs | (1, 9), (3, 3) |
Multiples vs Factors of 9
Factors of 9 are numbers that divide 9, but multiples of 9 are numbers you get by multiplying 9 by whole numbers. Example multiples: 9, 18, 27, 36, 45.
- Factors: always smaller or equal to 9.
- Multiples: always 9, or greater.
To know more, visit Multiples of 9.
Fast Trick: Divisibility Rule for 9
To check if a number is divisible by 9, add up all its digits. If the sum is a multiple of 9, the number is divisible by 9!
Example: Is 243 a multiple of 9?
2 + 4 + 3 = 9. Yes! 243 is divisible by 9.
Example Problems on Factors of 9
Solved Example 1: What are all the positive factors of 9?
1. Test each whole number up to 9.2. 9 ÷ 1 = 9 ✔️
3. 9 ÷ 2 = 4.5 (not whole)
4. 9 ÷ 3 = 3 ✔️
5. 9 ÷ 9 = 1 ✔️
Final Answer: 1, 3, 9
Solved Example 2: Find the sum of all the factors of 9.
1. List the factors: 1, 3, 92. Sum: 1 + 3 + 9 = 13
Final Answer: 13
Try These Yourself
- What are the factor pairs of 9?
- Is 27 a multiple or factor of 9?
- List the negative factors of 9.
- Are all factors of 9 factors of 81?
Revision Table: Factors of 9 from 1 to 20
| Number | Is Factor of 9? |
|---|---|
| 1 | Yes |
| 2 | No |
| 3 | Yes |
| 4 | No |
| 5 | No |
| 6 | No |
| 7 | No |
| 8 | No |
| 9 | Yes |
| 10 | No |
| 11 | No |
| 12 | No |
| 13 | No |
| 14 | No |
| 15 | No |
| 16 | No |
| 17 | No |
| 18 | No |
| 19 | No |
| 20 | No |
Relation to Other Number Concepts
Learning about factors of 9 builds a base for finding factors of 10, factors of 12, and even breaking down bigger numbers. It’s an essential step before tackling prime factorization and understanding factors and multiples. For more on the difference between prime and composite, read Prime and Composite Numbers.
Classroom Tip
A handy tip: If you know the square of 3 is 9, you can remember that 3 is repeated in the prime factorization of 9. Vedantu’s expert tutors show this visual cue when teaching factor trees and tables.
We’ve covered the factors of 9 from meaning, list, properties, tricks, and solved examples. You can learn more and practice similar concepts through Vedantu’s live online sessions and resources—for a confident maths journey!
FAQs on Factors of 9 Explained with Easy Steps & Examples
1. What are the factors of 9?
The factors of 9 are the whole numbers that divide 9 without leaving a remainder. These are 1, 3, and 9. We can also consider the negative factors: -1, -3, and -9.
2. How do I find the factors of 9 using division?
To find the factors, divide 9 by each whole number, starting from 1. If the division results in a whole number (no remainder), that number is a factor. For 9:
9 ÷ 1 = 9 (1 is a factor)
9 ÷ 2 = 4 with a remainder (2 is not a factor)
9 ÷ 3 = 3 (3 is a factor)
9 ÷ 4 = 2 with a remainder (4 is not a factor)
...and so on until you reach 9.
3. What is the prime factorization of 9?
Prime factorization breaks a number down into its prime factors (numbers only divisible by 1 and themselves). The prime factorization of 9 is 3 × 3 or 32.
4. What is a factor tree for 9?
A factor tree visually represents prime factorization. For 9, it would look like this:
This shows that 9 breaks down into 3 x 3.
5. Is 9 a prime or composite number?
9 is a composite number because it has more than two factors (1, 3, and 9).
6. What are the factor pairs of 9?
Factor pairs are sets of two numbers that multiply to give the original number. The factor pairs of 9 are (1, 9) and (3, 3). Including negative factors, we have (-1, -9) and (-3, -3).
7. What are some examples of using factors of 9?
Understanding factors helps in many areas, such as simplifying fractions (9/3 simplifies to 3), solving problems involving division, and understanding divisibility rules.
8. How many factors does 9 have?
Including both positive and negative factors, 9 has six factors: 1, 3, 9, -1, -3, and -9.
9. What is the difference between factors and multiples of 9?
Factors divide a number without leaving a remainder (1, 3, 9). Multiples are the results of multiplying a number by whole numbers (9, 18, 27, etc.).
10. What is the sum of the factors of 9?
The sum of the positive factors of 9 (1 + 3 + 9) is 13.
11. Are negative numbers considered factors?
Yes, negative numbers can also be factors. For example, -1, -3, and -9 are also factors of 9 because 9 divided by any of them results in a whole number.
12. How are factors of 9 used in higher-level math?
Factors are fundamental in algebra, number theory, and other advanced math concepts. For example, they're essential for finding greatest common divisors (GCD) and least common multiples (LCM).
