Q: What is the prime factorization of the number 2,877,579?

 A:
  • The prime factors are: 3 x 3 x 3 x 197 x 541
    • or also written as { 3, 3, 3, 197, 541 }
  • Written in exponential form: 33 x 1971 x 5411

Why is the prime factorization of 2,877,579 written as 33 x 1971 x 5411?

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 2,877,579

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 2,877,579 by 2

2,877,579 ÷ 2 = 1,438,789.5 - This has a remainder. Let's try another prime number.
2,877,579 ÷ 3 = 959,193 - No remainder! 3 is one of the factors!
959,193 ÷ 3 = 319,731 - No remainder! 3 is one of the factors!
319,731 ÷ 3 = 106,577 - No remainder! 3 is one of the factors!
106,577 ÷ 3 = 35,525.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
106,577 ÷ 5 = 21,315.4 - This has a remainder. 5 is not a factor.
106,577 ÷ 7 = 15,225.2857 - This has a remainder. 7 is not a factor.
106,577 ÷ 11 = 9,688.8182 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
106,577 ÷ 197 = 541 - No remainder! 197 is one of the factors!
541 ÷ 197 = 2.7462 - There is a remainder. We can't divide by 197 evenly anymore. Let's try the next prime number
541 ÷ 199 = 2.7186 - This has a remainder. 199 is not a factor.
541 ÷ 211 = 2.564 - This has a remainder. 211 is not a factor.
541 ÷ 223 = 2.426 - This has a remainder. 223 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
541 ÷ 541 = 1 - No remainder! 541 is one of the factors!

The orange divisor(s) above are the prime factors of the number 2,877,579. If we put all of it together we have the factors 3 x 3 x 3 x 197 x 541 = 2,877,579. It can also be written in exponential form as 33 x 1971 x 5411.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 2,877,579.

2,877,579
Factor Arrows
3959,193
Factor Arrows
3319,731
Factor Arrows
3106,577
Factor Arrows
197541

More Prime Factorization Examples

2,877,5772,877,5782,877,5802,877,581
1671 x 17,231121 x 111 x 1391 x 941122 x 51 x 143,879171 x 411,0831

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