Q: What is the prime factorization of the number 654,310,128?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 3 x 557 x 24,473
    • or also written as { 2, 2, 2, 2, 3, 557, 24,473 }
  • Written in exponential form: 24 x 31 x 5571 x 24,4731

Why is the prime factorization of 654,310,128 written as 24 x 31 x 5571 x 24,4731?

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 654,310,128

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 654,310,128 by 2

654,310,128 ÷ 2 = 327,155,064 - No remainder! 2 is one of the factors!
327,155,064 ÷ 2 = 163,577,532 - No remainder! 2 is one of the factors!
163,577,532 ÷ 2 = 81,788,766 - No remainder! 2 is one of the factors!
81,788,766 ÷ 2 = 40,894,383 - No remainder! 2 is one of the factors!
40,894,383 ÷ 2 = 20,447,191.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
40,894,383 ÷ 3 = 13,631,461 - No remainder! 3 is one of the factors!
13,631,461 ÷ 3 = 4,543,820.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
13,631,461 ÷ 5 = 2,726,292.2 - This has a remainder. 5 is not a factor.
13,631,461 ÷ 7 = 1,947,351.5714 - This has a remainder. 7 is not a factor.
13,631,461 ÷ 11 = 1,239,223.7273 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
13,631,461 ÷ 557 = 24,473 - No remainder! 557 is one of the factors!
24,473 ÷ 557 = 43.9372 - There is a remainder. We can't divide by 557 evenly anymore. Let's try the next prime number
24,473 ÷ 563 = 43.4689 - This has a remainder. 563 is not a factor.
24,473 ÷ 569 = 43.0105 - This has a remainder. 569 is not a factor.
24,473 ÷ 571 = 42.8599 - This has a remainder. 571 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
24,473 ÷ 24,473 = 1 - No remainder! 24,473 is one of the factors!

The orange divisor(s) above are the prime factors of the number 654,310,128. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 3 x 557 x 24,473 = 654,310,128. It can also be written in exponential form as 24 x 31 x 5571 x 24,4731.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 654,310,128.

654,310,128
Factor Arrows
2327,155,064
Factor Arrows
2163,577,532
Factor Arrows
281,788,766
Factor Arrows
240,894,383
Factor Arrows
313,631,461
Factor Arrows
55724,473

More Prime Factorization Examples

654,310,126654,310,127654,310,129654,310,130
21 x 327,155,0631171 x 38,488,8311111 x 59,482,739121 x 51 x 2,4371 x 26,8491

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