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 410,434,319 by 2
410,434,319 ÷ 2 = 205,217,159.5 - This has a remainder. Let's try another prime number.
410,434,319 ÷ 3 = 136,811,439.6667 - This has a remainder. Let's try another prime number.
410,434,319 ÷ 5 = 82,086,863.8 - This has a remainder. Let's try another prime number.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
410,434,319 ÷ 563 = 729,013 - No remainder! 563 is one of the factors!
729,013 ÷ 563 = 1,294.8721 - There is a remainder. We can't divide by 563 evenly anymore. Let's try the next prime number
729,013 ÷ 569 = 1,281.2179 - This has a remainder. 569 is not a factor.
729,013 ÷ 571 = 1,276.7303 - This has a remainder. 571 is not a factor.
729,013 ÷ 577 = 1,263.4541 - This has a remainder. 577 is not a factor.
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Keep trying increasingly larger numbers until you find one that divides evenly.
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729,013 ÷ 601 = 1,213 - No remainder! 601 is one of the factors!
1,213 ÷ 601 = 2.0183 - There is a remainder. We can't divide by 601 evenly anymore. Let's try the next prime number
1,213 ÷ 607 = 1.9984 - This has a remainder. 607 is not a factor.
1,213 ÷ 613 = 1.9788 - This has a remainder. 613 is not a factor.
1,213 ÷ 617 = 1.966 - This has a remainder. 617 is not a factor.
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Keep trying increasingly larger numbers until you find one that divides evenly.
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1,213 ÷ 1,213 = 1 - No remainder! 1,213 is one of the factors!
The orange divisor(s) above are the prime factors of the number 410,434,319. If we put all of it together we have the factors 563 x 601 x 1,213 = 410,434,319. It can also be written in exponential form as 5631 x 6011 x 1,2131.
Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 410,434,319.
| 410,434,319 | |||
 | |||
| 563 | 729,013 | ||
| |||
| 601 | 1,213 |
| 410,434,317 | 410,434,318 | 410,434,320 | 410,434,321 |
| 33 x 15,201,2711 | 21 x 71 x 3,3131 x 8,8491 | 24 x 31 x 51 x 8211 x 2,0831 | 111 x 2,2031 x 16,9371 |