Q: What is the prime factorization of the number 154,904,684?

 A:
  • The prime factors are: 2 x 2 x 11 x 11 x 271 x 1,181
    • or also written as { 2, 2, 11, 11, 271, 1,181 }
  • Written in exponential form: 22 x 112 x 2711 x 1,1811

Why is the prime factorization of 154,904,684 written as 22 x 112 x 2711 x 1,1811?

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 154,904,684

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 154,904,684 by 2

154,904,684 ÷ 2 = 77,452,342 - No remainder! 2 is one of the factors!
77,452,342 ÷ 2 = 38,726,171 - No remainder! 2 is one of the factors!
38,726,171 ÷ 2 = 19,363,085.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
38,726,171 ÷ 3 = 12,908,723.6667 - This has a remainder. 3 is not a factor.
38,726,171 ÷ 5 = 7,745,234.2 - This has a remainder. 5 is not a factor.
38,726,171 ÷ 7 = 5,532,310.1429 - This has a remainder. 7 is not a factor.
38,726,171 ÷ 11 = 3,520,561 - No remainder! 11 is one of the factors!
3,520,561 ÷ 11 = 320,051 - No remainder! 11 is one of the factors!
320,051 ÷ 11 = 29,095.5455 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
320,051 ÷ 13 = 24,619.3077 - This has a remainder. 13 is not a factor.
320,051 ÷ 17 = 18,826.5294 - This has a remainder. 17 is not a factor.
320,051 ÷ 19 = 16,844.7895 - This has a remainder. 19 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
320,051 ÷ 271 = 1,181 - No remainder! 271 is one of the factors!
1,181 ÷ 271 = 4.3579 - There is a remainder. We can't divide by 271 evenly anymore. Let's try the next prime number
1,181 ÷ 277 = 4.2635 - This has a remainder. 277 is not a factor.
1,181 ÷ 281 = 4.2028 - This has a remainder. 281 is not a factor.
1,181 ÷ 283 = 4.1731 - This has a remainder. 283 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,181 ÷ 1,181 = 1 - No remainder! 1,181 is one of the factors!

The orange divisor(s) above are the prime factors of the number 154,904,684. If we put all of it together we have the factors 2 x 2 x 11 x 11 x 271 x 1,181 = 154,904,684. It can also be written in exponential form as 22 x 112 x 2711 x 1,1811.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 154,904,684.

154,904,684
Factor Arrows
277,452,342
Factor Arrows
238,726,171
Factor Arrows
113,520,561
Factor Arrows
11320,051
Factor Arrows
2711,181

More Prime Factorization Examples

154,904,682154,904,683154,904,685154,904,686
21 x 31 x 191 x 1,358,8131411 x 3,778,163131 x 51 x 131 x 794,383121 x 731 x 1,060,9911

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