Factors of 9 | Prime Factorization of 9 - Explained Simply

Today we are going to present here Factor Tree of 9. The factor is the number that divides the original number. The factors of 9 are 1, 3 and 9 itself.

What are the factors of 9?

We want to find all the whole numbers that can divide 9 without leaving any remainder.

Factor Tree Method of 9: Explained Simply

Finding factors is a systematic process. For a small number like 9, it's quite straightforward. We'll check numbers starting from 1 and see if they divide 9 evenly.

Here’s how to find the factors of 9:

Start with 1:

Is 1 a factor of 9? Yes, always! Any number divided by 1 is itself.

1 × 9 = 9

So, 1 and 9 are factors.

Test the next whole number, 2:

Can 9 be divided by 2 evenly? 9 ÷ 2 = 4 with a remainder of 1.

No, 2 is not a factor of 9.

Test the next whole number, 3:

Can 9 be divided by 3 evenly? Yes! 9 ÷ 3 = 3 with no remainder.

3 × 3 = 9

So, 3 is a factor. In this case, it appears twice in the multiplication (3 and 3), but we only list it once in the set of factors.

Continue testing (optional, but good for understanding when to stop):

The general rule is that you only need to test numbers up to the square root of the number you're factoring (which for 9 is 3, since 3 x 3 = 9). Once you reach a factor that is repeated (like 3 in 3x3=9), or you pass the square root, you've found all the unique factors.

However, for clarity, let's briefly check numbers beyond 3:

Test 4: 9 ÷ 4 = 2 remainder 1. (No)

Test 5: 9 ÷ 5 = 1 remainder 4. (No)

Test 6: 9 ÷ 6 = 1 remainder 3. (No)

Test 7: 9 ÷ 7 = 1 remainder 2. (No)

Test 8: 9 ÷ 8 = 1 remainder 1. (No)

We already found 9 itself as a factor with 1.

The Factors Pairs of 9

The positive factors of 9 are 1, 3 and 9.

We can also list these as factor pairs:

(1, 9) because 1 × 9 = 9

(3, 3) because 3 × 3 = 9 (note: we only list 3 once in the final set of factors)

Important Note on Negative Factors:

While in basic math we usually focus on positive factors, it's good to remember that negative numbers can also be factors.

(-1) × (-9) = 9

(-3) × (-3) = 9

So, the complete set of integer factors for 9 would be: -9, -3, -1, 1, 3, 9. For most contexts, "factors" refers to the positive ones.

Prime Factorization of 9

There are primarily two popular methods for prime factorization:

First, let's identify 9:

Is 9 a prime number? No, because it has factors 1, 3 and 9 (more than two factors).

Therefore, 9 is a composite number.

Now, let's find its prime factorization:

Factor Tree Method

/ \

3 3

Start with 9.

The factors of 9 are 1, 3, 9. We can break 9 down into 3 × 3.

Both 3s are prime numbers. We circle them.

All branches end in prime numbers.

Therefore, the prime factorization of 9 is 3 × 3.

This can also be written in exponential form as 3².

The Division Method

3 | 9

3 | 3

Start with 9.

The smallest prime number that divides 9 evenly is 3 (9 ÷ 3 = 3).

Write 3 on the left, and 3 below 9.

Now we work with the new number, 3. The smallest prime number that divides 3 evenly is 3 (3 ÷ 3 = 1).

Write 3 on the left and 1 below 3.

We stop when the final result is 1.

The prime divisors are the numbers on the left: 3 and 3.

Therefore, the prime factorization of 9 is 3 × 3.

Again, this can be written as 3².

Conclusion

To simplify fractions, you can find the prime factorization of the numerator and the denominator, then cancel out common prime factors.