Q: What is the prime factorization of the number 1,653,648?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 3 x 47 x 733
    • or also written as { 2, 2, 2, 2, 3, 47, 733 }
  • Written in exponential form: 24 x 31 x 471 x 7331

Why is the prime factorization of 1,653,648 written as 24 x 31 x 471 x 7331?

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 1,653,648

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 1,653,648 by 2

1,653,648 ÷ 2 = 826,824 - No remainder! 2 is one of the factors!
826,824 ÷ 2 = 413,412 - No remainder! 2 is one of the factors!
413,412 ÷ 2 = 206,706 - No remainder! 2 is one of the factors!
206,706 ÷ 2 = 103,353 - No remainder! 2 is one of the factors!
103,353 ÷ 2 = 51,676.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
103,353 ÷ 3 = 34,451 - No remainder! 3 is one of the factors!
34,451 ÷ 3 = 11,483.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
34,451 ÷ 5 = 6,890.2 - This has a remainder. 5 is not a factor.
34,451 ÷ 7 = 4,921.5714 - This has a remainder. 7 is not a factor.
34,451 ÷ 11 = 3,131.9091 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
34,451 ÷ 47 = 733 - No remainder! 47 is one of the factors!
733 ÷ 47 = 15.5957 - There is a remainder. We can't divide by 47 evenly anymore. Let's try the next prime number
733 ÷ 53 = 13.8302 - This has a remainder. 53 is not a factor.
733 ÷ 59 = 12.4237 - This has a remainder. 59 is not a factor.
733 ÷ 61 = 12.0164 - This has a remainder. 61 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
733 ÷ 733 = 1 - No remainder! 733 is one of the factors!

The orange divisor(s) above are the prime factors of the number 1,653,648. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 3 x 47 x 733 = 1,653,648. It can also be written in exponential form as 24 x 31 x 471 x 7331.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 1,653,648.

1,653,648
Factor Arrows
2826,824
Factor Arrows
2413,412
Factor Arrows
2206,706
Factor Arrows
2103,353
Factor Arrows
334,451
Factor Arrows
47733

More Prime Factorization Examples

1,653,6461,653,6471,653,6491,653,650
21 x 191 x 43,51714671 x 3,5411611 x 27,109121 x 52 x 33,0731

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