Q: What is the prime factorization of the number 154,835,200?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5 x 13 x 1,861
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 5, 5, 13, 1,861 }
  • Written in exponential form: 28 x 52 x 131 x 1,8611

Why is the prime factorization of 154,835,200 written as 28 x 52 x 131 x 1,8611?

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 154,835,200

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 154,835,200 by 2

154,835,200 ÷ 2 = 77,417,600 - No remainder! 2 is one of the factors!
77,417,600 ÷ 2 = 38,708,800 - No remainder! 2 is one of the factors!
38,708,800 ÷ 2 = 19,354,400 - No remainder! 2 is one of the factors!
19,354,400 ÷ 2 = 9,677,200 - No remainder! 2 is one of the factors!
9,677,200 ÷ 2 = 4,838,600 - No remainder! 2 is one of the factors!
4,838,600 ÷ 2 = 2,419,300 - No remainder! 2 is one of the factors!
2,419,300 ÷ 2 = 1,209,650 - No remainder! 2 is one of the factors!
1,209,650 ÷ 2 = 604,825 - No remainder! 2 is one of the factors!
604,825 ÷ 2 = 302,412.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
604,825 ÷ 3 = 201,608.3333 - This has a remainder. 3 is not a factor.
604,825 ÷ 5 = 120,965 - No remainder! 5 is one of the factors!
120,965 ÷ 5 = 24,193 - No remainder! 5 is one of the factors!
24,193 ÷ 5 = 4,838.6 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
24,193 ÷ 7 = 3,456.1429 - This has a remainder. 7 is not a factor.
24,193 ÷ 11 = 2,199.3636 - This has a remainder. 11 is not a factor.
24,193 ÷ 13 = 1,861 - No remainder! 13 is one of the factors!
1,861 ÷ 13 = 143.1538 - There is a remainder. We can't divide by 13 evenly anymore. Let's try the next prime number
1,861 ÷ 17 = 109.4706 - This has a remainder. 17 is not a factor.
1,861 ÷ 19 = 97.9474 - This has a remainder. 19 is not a factor.
1,861 ÷ 23 = 80.913 - This has a remainder. 23 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,861 ÷ 1,861 = 1 - No remainder! 1,861 is one of the factors!

The orange divisor(s) above are the prime factors of the number 154,835,200. 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 2 x 5 x 5 x 13 x 1,861 = 154,835,200. It can also be written in exponential form as 28 x 52 x 131 x 1,8611.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 154,835,200.

154,835,200
Factor Arrows
277,417,600
Factor Arrows
238,708,800
Factor Arrows
219,354,400
Factor Arrows
29,677,200
Factor Arrows
24,838,600
Factor Arrows
22,419,300
Factor Arrows
21,209,650
Factor Arrows
2604,825
Factor Arrows
5120,965
Factor Arrows
524,193
Factor Arrows
131,861

More Prime Factorization Examples

154,835,198154,835,199154,835,201154,835,202
21 x 72 x 1,579,951133 x 191 x 1131 x 2,6711171 x 2631 x 34,631121 x 31 x 471 x 611 x 9,0011

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