Q: What is the prime factorization of the number 289,760?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 5 x 1,811
    • or also written as { 2, 2, 2, 2, 2, 5, 1,811 }
  • Written in exponential form: 25 x 51 x 1,8111

Why is the prime factorization of 289,760 written as 25 x 51 x 1,8111?

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 289,760

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 289,760 by 2

289,760 ÷ 2 = 144,880 - No remainder! 2 is one of the factors!
144,880 ÷ 2 = 72,440 - No remainder! 2 is one of the factors!
72,440 ÷ 2 = 36,220 - No remainder! 2 is one of the factors!
36,220 ÷ 2 = 18,110 - No remainder! 2 is one of the factors!
18,110 ÷ 2 = 9,055 - No remainder! 2 is one of the factors!
9,055 ÷ 2 = 4,527.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
9,055 ÷ 3 = 3,018.3333 - This has a remainder. 3 is not a factor.
9,055 ÷ 5 = 1,811 - No remainder! 5 is one of the factors!
1,811 ÷ 5 = 362.2 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
1,811 ÷ 7 = 258.7143 - This has a remainder. 7 is not a factor.
1,811 ÷ 11 = 164.6364 - This has a remainder. 11 is not a factor.
1,811 ÷ 13 = 139.3077 - This has a remainder. 13 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
1,811 ÷ 1,811 = 1 - No remainder! 1,811 is one of the factors!

The orange divisor(s) above are the prime factors of the number 289,760. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 5 x 1,811 = 289,760. It can also be written in exponential form as 25 x 51 x 1,8111.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 289,760.

289,760
Factor Arrows
2144,880
Factor Arrows
272,440
Factor Arrows
236,220
Factor Arrows
218,110
Factor Arrows
29,055
Factor Arrows
51,811

More Prime Factorization Examples

289,758289,759289,761289,762
21 x 31 x 71 x 6,8991289,759131 x 96,587121 x 111 x 13,1711

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