Q: What is the prime factorization of the number 35,140,224?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 7 x 17 x 769
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 3, 7, 17, 769 }
  • Written in exponential form: 27 x 31 x 71 x 171 x 7691

Why is the prime factorization of 35,140,224 written as 27 x 31 x 71 x 171 x 7691?

What is prime factorization?

Prime factorization or prime factor decomposition is the process of finding which prime numbers can be multiplied together to make the original number.

Finding the prime factors of 35,140,224

To find the prime factors, you start by dividing the number by the first prime number, which is 2. If there is not a remainder, meaning you can divide evenly, then 2 is a factor of the number. Continue dividing by 2 until you cannot divide evenly anymore. Write down how many 2's you were able to divide by evenly. Now try dividing by the next prime factor, which is 3. The goal is to get to a quotient of 1.

If it doesn't make sense yet, let's try it...

Here are the first several prime factors: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...

Let's start by dividing 35,140,224 by 2

35,140,224 ÷ 2 = 17,570,112 - No remainder! 2 is one of the factors!
17,570,112 ÷ 2 = 8,785,056 - No remainder! 2 is one of the factors!
8,785,056 ÷ 2 = 4,392,528 - No remainder! 2 is one of the factors!
4,392,528 ÷ 2 = 2,196,264 - No remainder! 2 is one of the factors!
2,196,264 ÷ 2 = 1,098,132 - No remainder! 2 is one of the factors!
1,098,132 ÷ 2 = 549,066 - No remainder! 2 is one of the factors!
549,066 ÷ 2 = 274,533 - No remainder! 2 is one of the factors!
274,533 ÷ 2 = 137,266.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
274,533 ÷ 3 = 91,511 - No remainder! 3 is one of the factors!
91,511 ÷ 3 = 30,503.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
91,511 ÷ 5 = 18,302.2 - This has a remainder. 5 is not a factor.
91,511 ÷ 7 = 13,073 - No remainder! 7 is one of the factors!
13,073 ÷ 7 = 1,867.5714 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
13,073 ÷ 11 = 1,188.4545 - This has a remainder. 11 is not a factor.
13,073 ÷ 13 = 1,005.6154 - This has a remainder. 13 is not a factor.
13,073 ÷ 17 = 769 - No remainder! 17 is one of the factors!
769 ÷ 17 = 45.2353 - There is a remainder. We can't divide by 17 evenly anymore. Let's try the next prime number
769 ÷ 19 = 40.4737 - This has a remainder. 19 is not a factor.
769 ÷ 23 = 33.4348 - This has a remainder. 23 is not a factor.
769 ÷ 29 = 26.5172 - This has a remainder. 29 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
769 ÷ 769 = 1 - No remainder! 769 is one of the factors!

The orange divisor(s) above are the prime factors of the number 35,140,224. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 7 x 17 x 769 = 35,140,224. It can also be written in exponential form as 27 x 31 x 71 x 171 x 7691.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 35,140,224.

35,140,224
Factor Arrows
217,570,112
Factor Arrows
28,785,056
Factor Arrows
24,392,528
Factor Arrows
22,196,264
Factor Arrows
21,098,132
Factor Arrows
2549,066
Factor Arrows
2274,533
Factor Arrows
391,511
Factor Arrows
713,073
Factor Arrows
17769

More Prime Factorization Examples

35,140,22235,140,22335,140,22535,140,226
21 x 131 x 1,351,5471591 x 1011 x 5,897152 x 6131 x 2,293121 x 111 x 891 x 1311 x 1371

Try the factor calculator

Explore more about the number 35,140,224:


Ask a Question