Listing All Factors: Discover Every Divisor of 720

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

All factors include every number that divides 720 evenly (1, 2, 3, 4, 5, 6, 8, 9, 10...). Prime factors are only the prime numbers that multiply together to make 720.

Q: Why do I need to repeat the same prime number?

Because some prime numbers divide the original number multiple times! For 720, we get four 2's because $ 720 = 2 × 2 × 2 × 2 × 3 × 3 × 5 $.

Q: Do I need to write the factors in order?

The order doesn't matter mathematically, but it's conventional to write them from smallest to largest or list all of one prime before moving to the next.

Q: What if I make a mistake in my divisions?

Always check your work by multiplying all your prime factors together. If you don't get the original number (720), go back and find your error!

Q: Is there a faster way than dividing step by step?

You can use a factor tree! Start with 720, split it into any two factors, then keep splitting until all branches end in prime numbers.

Prime Factorization with Multiple Repeated Factors

Write all the factors of the following number: $720$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find all the prime factors of the number

00:04 The ones digit is 0, therefore 2 is definitely a prime factor

00:13 Divide by 2, and continue with the result to find the factors

00:20 The ones digit is 0, therefore 2 is definitely a prime factor

00:25 Divide by 2, and continue with the result to find the factors

00:31 The ones digit is 0, therefore 2 is definitely a prime factor

00:35 Divide by 2, and continue with the result to find the factors

00:38 The ones digit is 0, therefore 2 is definitely a prime factor

00:41 Divide by 2, and continue with the result to find the factors

00:44 The ones digit is 5, therefore 5 is definitely a prime factor

00:48 Divide by 5, and continue with the result to find the factors

00:53 The ones digit is 9, therefore 3 is definitely a prime factor

00:56 Divide by 3, and continue with the result to find the factors

00:59 And the result is a prime number, therefore it is a factor by itself

01:02 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Write all the factors of the following number: $720$

Step-by-step solution

To write all the factors of the number 720 using prime factorization, we will proceed as follows:

Step 1: Start with the smallest prime number, 2, and divide 720 by 2.

$720 \div 2 = 360$

Divide by 2 again.

$360 \div 2 = 180$

Continue dividing by 2 until it no longer divides evenly.

$180 \div 2 = 90$

$90 \div 2 = 45$

Step 2: Since 45 is not divisible by 2, move to the next prime number, which is 3.

$45 \div 3 = 15$

Continue dividing by 3.

$15 \div 3 = 5$

Step 3: Now divide by the next prime number, 5.

$5 \div 5 = 1$

The prime factorization of 720 is therefore:

$720 = 2^4 \times 3^2 \times 5^1$

Thus, the factors (including multiplicities) are:

$5,3,3,2,2,2,2$

Therefore, the solution to the problem is $5,3,3,2,2,2,2$ .

Final Answer

$5,3,3,2,2,2,2$

Key Points to Remember

Essential concepts to master this topic

Method: Divide by smallest prime repeatedly until no longer divisible
Technique: $720 ÷ 2 = 360 ÷ 2 = 180 ÷ 2 = 90 ÷ 2 = 45$
Check: Multiply all prime factors: $2^4 × 3^2 × 5 = 16 × 9 × 5 = 720$ ✓

Common Mistakes

Avoid these frequent errors

Listing all factors instead of just prime factors
Don't list every factor like 1, 2, 3, 4, 5, 6, 8, 9, 10... = wrong approach! The question asks for prime factorization, not all divisors. Always break the number down using only prime numbers (2, 3, 5, 7, 11...).

Practice Quiz

Test your knowledge with interactive questions

Write all the factors of the following number: $$ 6 $$

FAQ

Everything you need to know about this question

What's the difference between factors and prime factors?

All factors include every number that divides 720 evenly (1, 2, 3, 4, 5, 6, 8, 9, 10...). Prime factors are only the prime numbers that multiply together to make 720.

Why do I need to repeat the same prime number?

Because some prime numbers divide the original number multiple times! For 720, we get four 2's because $720 = 2 × 2 × 2 × 2 × 3 × 3 × 5$ .

How do I know when to stop dividing?

Stop when you reach 1. If your last division gives you a prime number (like 5), divide by that prime once more to get 1.

Do I need to write the factors in order?

The order doesn't matter mathematically, but it's conventional to write them from smallest to largest or list all of one prime before moving to the next.

What if I make a mistake in my divisions?

Always check your work by multiplying all your prime factors together. If you don't get the original number (720), go back and find your error!

Is there a faster way than dividing step by step?

You can use a factor tree! Start with 720, split it into any two factors, then keep splitting until all branches end in prime numbers.

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