Q: What is the prime factorization of the number 35,375,360?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5 x 29 x 953
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 5, 29, 953 }
  • Written in exponential form: 28 x 51 x 291 x 9531

Why is the prime factorization of 35,375,360 written as 28 x 51 x 291 x 9531?

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 35,375,360

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 35,375,360 by 2

35,375,360 ÷ 2 = 17,687,680 - No remainder! 2 is one of the factors!
17,687,680 ÷ 2 = 8,843,840 - No remainder! 2 is one of the factors!
8,843,840 ÷ 2 = 4,421,920 - No remainder! 2 is one of the factors!
4,421,920 ÷ 2 = 2,210,960 - No remainder! 2 is one of the factors!
2,210,960 ÷ 2 = 1,105,480 - No remainder! 2 is one of the factors!
1,105,480 ÷ 2 = 552,740 - No remainder! 2 is one of the factors!
552,740 ÷ 2 = 276,370 - No remainder! 2 is one of the factors!
276,370 ÷ 2 = 138,185 - No remainder! 2 is one of the factors!
138,185 ÷ 2 = 69,092.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
138,185 ÷ 3 = 46,061.6667 - This has a remainder. 3 is not a factor.
138,185 ÷ 5 = 27,637 - No remainder! 5 is one of the factors!
27,637 ÷ 5 = 5,527.4 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
27,637 ÷ 7 = 3,948.1429 - This has a remainder. 7 is not a factor.
27,637 ÷ 11 = 2,512.4545 - This has a remainder. 11 is not a factor.
27,637 ÷ 13 = 2,125.9231 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
27,637 ÷ 29 = 953 - No remainder! 29 is one of the factors!
953 ÷ 29 = 32.8621 - There is a remainder. We can't divide by 29 evenly anymore. Let's try the next prime number
953 ÷ 31 = 30.7419 - This has a remainder. 31 is not a factor.
953 ÷ 37 = 25.7568 - This has a remainder. 37 is not a factor.
953 ÷ 41 = 23.2439 - This has a remainder. 41 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
953 ÷ 953 = 1 - No remainder! 953 is one of the factors!

The orange divisor(s) above are the prime factors of the number 35,375,360. 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 29 x 953 = 35,375,360. It can also be written in exponential form as 28 x 51 x 291 x 9531.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 35,375,360.

35,375,360
Factor Arrows
217,687,680
Factor Arrows
28,843,840
Factor Arrows
24,421,920
Factor Arrows
22,210,960
Factor Arrows
21,105,480
Factor Arrows
2552,740
Factor Arrows
2276,370
Factor Arrows
2138,185
Factor Arrows
527,637
Factor Arrows
29953

More Prime Factorization Examples

35,375,35835,375,35935,375,36135,375,362
21 x 31 x 5,895,8931191 x 1,861,861131 x 71 x 1391 x 12,119121 x 111 x 731 x 22,0271

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