Prime factorization or prime factor decomposition is the process of finding which prime numbers can be multiplied together to make the original number.
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.
Here are the first several prime factors: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...
Let's start by dividing 220,221,011 by 2
220,221,011 ÷ 2 = 110,110,505.5 - This has a remainder. Let's try another prime number.
220,221,011 ÷ 3 = 73,407,003.6667 - This has a remainder. Let's try another prime number.
220,221,011 ÷ 5 = 44,044,202.2 - This has a remainder. Let's try another prime number.
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Keep trying increasingly larger numbers until you find one that divides evenly.
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220,221,011 ÷ 359 = 613,429 - No remainder! 359 is one of the factors!
613,429 ÷ 359 = 1,708.7159 - There is a remainder. We can't divide by 359 evenly anymore. Let's try the next prime number
613,429 ÷ 367 = 1,671.4687 - This has a remainder. 367 is not a factor.
613,429 ÷ 373 = 1,644.5818 - This has a remainder. 373 is not a factor.
613,429 ÷ 379 = 1,618.5462 - This has a remainder. 379 is not a factor.
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Keep trying increasingly larger numbers until you find one that divides evenly.
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613,429 ÷ 619 = 991 - No remainder! 619 is one of the factors!
991 ÷ 619 = 1.601 - There is a remainder. We can't divide by 619 evenly anymore. Let's try the next prime number
991 ÷ 631 = 1.5705 - This has a remainder. 631 is not a factor.
991 ÷ 641 = 1.546 - This has a remainder. 641 is not a factor.
991 ÷ 643 = 1.5412 - This has a remainder. 643 is not a factor.
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Keep trying increasingly larger numbers until you find one that divides evenly.
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991 ÷ 991 = 1 - No remainder! 991 is one of the factors!
The orange divisor(s) above are the prime factors of the number 220,221,011. If we put all of it together we have the factors 359 x 619 x 991 = 220,221,011. It can also be written in exponential form as 3591 x 6191 x 9911.
Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 220,221,011.
| 220,221,011 | |||
 | |||
| 359 | 613,429 | ||
| |||
| 619 | 991 |
| 220,221,009 | 220,221,010 | 220,221,012 | 220,221,013 |
| 32 x 171 x 6911 x 2,0831 | 21 x 51 x 22,022,1011 | 22 x 31 x 111 x 291 x 57,5291 | 14,5371 x 15,1491 |