Q: What is the prime factorization of the number 3,223,314?

 A:
  • The prime factors are: 2 x 3 x 3 x 3 x 3 x 101 x 197
    • or also written as { 2, 3, 3, 3, 3, 101, 197 }
  • Written in exponential form: 21 x 34 x 1011 x 1971

Why is the prime factorization of 3,223,314 written as 21 x 34 x 1011 x 1971?

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 3,223,314

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 3,223,314 by 2

3,223,314 ÷ 2 = 1,611,657 - No remainder! 2 is one of the factors!
1,611,657 ÷ 2 = 805,828.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
1,611,657 ÷ 3 = 537,219 - No remainder! 3 is one of the factors!
537,219 ÷ 3 = 179,073 - No remainder! 3 is one of the factors!
179,073 ÷ 3 = 59,691 - No remainder! 3 is one of the factors!
59,691 ÷ 3 = 19,897 - No remainder! 3 is one of the factors!
19,897 ÷ 3 = 6,632.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
19,897 ÷ 5 = 3,979.4 - This has a remainder. 5 is not a factor.
19,897 ÷ 7 = 2,842.4286 - This has a remainder. 7 is not a factor.
19,897 ÷ 11 = 1,808.8182 - This has a remainder. 11 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
19,897 ÷ 101 = 197 - No remainder! 101 is one of the factors!
197 ÷ 101 = 1.9505 - There is a remainder. We can't divide by 101 evenly anymore. Let's try the next prime number
197 ÷ 103 = 1.9126 - This has a remainder. 103 is not a factor.
197 ÷ 107 = 1.8411 - This has a remainder. 107 is not a factor.
197 ÷ 109 = 1.8073 - This has a remainder. 109 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
197 ÷ 197 = 1 - No remainder! 197 is one of the factors!

The orange divisor(s) above are the prime factors of the number 3,223,314. If we put all of it together we have the factors 2 x 3 x 3 x 3 x 3 x 101 x 197 = 3,223,314. It can also be written in exponential form as 21 x 34 x 1011 x 1971.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 3,223,314.

3,223,314
Factor Arrows
21,611,657
Factor Arrows
3537,219
Factor Arrows
3179,073
Factor Arrows
359,691
Factor Arrows
319,897
Factor Arrows
101197

More Prime Factorization Examples

3,223,3123,223,3133,223,3153,223,316
24 x 191 x 231 x 4611891 x 36,217151 x 731 x 8,831122 x 1911 x 4,2191

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