Q: What is the prime factorization of the number 34,542,000?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 3 x 3 x 5 x 5 x 5 x 19 x 101
    • or also written as { 2, 2, 2, 2, 3, 3, 5, 5, 5, 19, 101 }
  • Written in exponential form: 24 x 32 x 53 x 191 x 1011

Why is the prime factorization of 34,542,000 written as 24 x 32 x 53 x 191 x 1011?

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 34,542,000

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 34,542,000 by 2

34,542,000 ÷ 2 = 17,271,000 - No remainder! 2 is one of the factors!
17,271,000 ÷ 2 = 8,635,500 - No remainder! 2 is one of the factors!
8,635,500 ÷ 2 = 4,317,750 - No remainder! 2 is one of the factors!
4,317,750 ÷ 2 = 2,158,875 - No remainder! 2 is one of the factors!
2,158,875 ÷ 2 = 1,079,437.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
2,158,875 ÷ 3 = 719,625 - No remainder! 3 is one of the factors!
719,625 ÷ 3 = 239,875 - No remainder! 3 is one of the factors!
239,875 ÷ 3 = 79,958.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
239,875 ÷ 5 = 47,975 - No remainder! 5 is one of the factors!
47,975 ÷ 5 = 9,595 - No remainder! 5 is one of the factors!
9,595 ÷ 5 = 1,919 - No remainder! 5 is one of the factors!
1,919 ÷ 5 = 383.8 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
1,919 ÷ 7 = 274.1429 - This has a remainder. 7 is not a factor.
1,919 ÷ 11 = 174.4545 - This has a remainder. 11 is not a factor.
1,919 ÷ 13 = 147.6154 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,919 ÷ 19 = 101 - No remainder! 19 is one of the factors!
101 ÷ 19 = 5.3158 - There is a remainder. We can't divide by 19 evenly anymore. Let's try the next prime number
101 ÷ 23 = 4.3913 - This has a remainder. 23 is not a factor.
101 ÷ 29 = 3.4828 - This has a remainder. 29 is not a factor.
101 ÷ 31 = 3.2581 - This has a remainder. 31 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
101 ÷ 101 = 1 - No remainder! 101 is one of the factors!

The orange divisor(s) above are the prime factors of the number 34,542,000. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 3 x 3 x 5 x 5 x 5 x 19 x 101 = 34,542,000. It can also be written in exponential form as 24 x 32 x 53 x 191 x 1011.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 34,542,000.

34,542,000
Factor Arrows
217,271,000
Factor Arrows
28,635,500
Factor Arrows
24,317,750
Factor Arrows
22,158,875
Factor Arrows
3719,625
Factor Arrows
3239,875
Factor Arrows
547,975
Factor Arrows
59,595
Factor Arrows
51,919
Factor Arrows
19101

More Prime Factorization Examples

34,541,99834,541,99934,542,00134,542,002
21 x 231 x 311 x 24,223134,541,9991131 x 2,657,077121 x 111 x 1,570,0911

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