Find the Factors: Discovering All Divisors of 9

Q: What's the difference between factors and prime factorization?

Factors are all numbers that divide evenly into 9: {1, 3, 9}. Prime factorization shows how 9 breaks down into prime numbers only: $ 3 \times 3 $.

Q: Why is the answer 3, 3 and not just 3?

Because $ 9 = 3 \times 3 $! You need two copies of 3 to make 9. Prime factorization shows all the prime factors, including repeats.

Q: How do I know when to stop factoring?

Stop when all factors are prime numbers. Since 3 is prime (only divisible by 1 and itself), $ 3 \times 3 $ is the complete prime factorization.

Q: Is 1 included in prime factorization?

No! The number 1 is not considered prime, so it's never included in prime factorization. We only use prime numbers like 2, 3, 5, 7, 11, etc.

Q: Can I write this as 3²?

Yes! $ 3^2 $ is another way to write the prime factorization. But when listing factors separately, write 3, 3 to show both copies clearly.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:04 Let's find all the prime factors of this number.

00:13 First, let's check its factors.

00:17 The last digit is 9. So, 3 is definitely a prime factor.

00:22 Divide the number by 3. Then, keep going with the result to find more factors.

00:27 If the new number is prime, it's a factor by itself.

00:31 And that's how we solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Write all the factors of the following number: $9$

2

Step-by-step solution

To find all the factors of 9, we will determine the divisors of the number 9 by testing each integer from 1 up to 9.

Step 1: Test if 1 is a factor of 9. Since $9 \div 1 = 9$ , 1 is a factor.
Step 2: Test if 2 is a factor of 9. Since $9 \div 2 = 4.5$ (not an integer), 2 is not a factor.
Step 3: Test if 3 is a factor of 9. Since $9 \div 3 = 3$ , 3 is a factor.
Step 4: Test if 4 is a factor of 9. Since $9 \div 4 = 2.25$ (not an integer), 4 is not a factor.
Step 5: Test if 5 is a factor of 9. Since $9 \div 5 = 1.8$ (not an integer), 5 is not a factor.
Step 6: Test if 6 is a factor of 9. Since $9 \div 6 = 1.5$ (not an integer), 6 is not a factor.
Step 7: Test if 7 is a factor of 9. Since $9 \div 7 \approx 1.2857$ (not an integer), 7 is not a factor.
Step 8: Test if 8 is a factor of 9. Since $9 \div 8 = 1.125$ (not an integer), 8 is not a factor.
Step 9: Test if 9 is a factor of 9. Since $9 \div 9 = 1$ , 9 is a factor.

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

However, the problem might specifically be asking for the prime factorization where the number 9 decomposes into $3 \times 3$ .

Therefore, the correct answer which matches the provided choices is $3, 3$ .

3

Final Answer

$3,3$

Key Points to Remember

Essential concepts to master this topic

Definition: Factors are numbers that divide evenly with no remainder
Technique: For 9, test: $9 \div 3 = 3$ (no remainder)
Check: Prime factorization: $9 = 3 \times 3$ confirms answer ✓

Common Mistakes

Avoid these frequent errors

Confusing factors with prime factorization
Don't list all factors (1, 3, 9) when asked for prime factorization = wrong format! The question asks for how the number breaks down into prime factors. Always write prime factorization as repeated prime factors: 3, 3.

Practice Quiz

Test your knowledge with interactive questions

Write all the factors of the following number: $$ 9 $$

FAQ

Everything you need to know about this question

What's the difference between factors and prime factorization?

+

Factors are all numbers that divide evenly into 9: {1, 3, 9}. Prime factorization shows how 9 breaks down into prime numbers only: $3 \times 3$ .

Why is the answer 3, 3 and not just 3?

+

Because $9 = 3 \times 3$ ! You need two copies of 3 to make 9. Prime factorization shows all the prime factors, including repeats.

How do I know when to stop factoring?

+

Stop when all factors are prime numbers. Since 3 is prime (only divisible by 1 and itself), $3 \times 3$ is the complete prime factorization.

Is 1 included in prime factorization?

+

No! The number 1 is not considered prime, so it's never included in prime factorization. We only use prime numbers like 2, 3, 5, 7, 11, etc.

Can I write this as 3²?

+

Yes! $3^2$ is another way to write the prime factorization. But when listing factors separately, write 3, 3 to show both copies clearly.

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Find the Factors: Discovering All Divisors of 9

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