How to Find the Prime Factorization of a Number?

Prime factorization involves breaking down a number into a product of its prime factors. Here's how you do it:

The Basic Method: Trial Division

This is the most straightforward method.

1. Start with the smallest prime number, 2.
2. Divide the number by 2. If it divides evenly (remainder 0), 2 is a prime factor. Record it.
3. Repeat step 2 with the result of the division until it is no longer divisible by 2.
4. Move to the next prime number, 3.
5. Divide the current result by 3. If it divides evenly, 3 is a prime factor. Record it.
6. Repeat step 5 with the result of the division until it is no longer divisible by 3.
7. Continue this process with the next prime numbers (5, 7, 11, 13, etc.) until the result of the division is 1.
8. The prime factors you recorded are the prime factorization of the original number.

Example: Prime Factorization of 30

• 30 / 2 = 15 (2 is a prime factor)
• 15 / 3 = 5 (3 is a prime factor)
• 5 / 5 = 1 (5 is a prime factor)

Therefore, the prime factorization of 30 is 2 x 3 x 5.

An Optimized Method:

This optimization reduces the number of divisions you need to perform.

1. Divide by 2 until you can't. This is the same as the first few steps of the basic method.
2. After dividing by 2 as many times as possible, only check odd numbers as potential prime factors. You can skip all even numbers because if the original number were divisible by any even number greater than 2, it would have already been divisible by 2 during the first step.
3. Only check potential factors up to the square root of the number. If a number n has a prime factor greater than its square root, it must also have a prime factor smaller than its square root. This dramatically reduces the range of numbers you need to check. For instance, if you're finding the prime factors of 100, you only need to check up to 10 because the square root of 100 is 10.

Example: Prime Factorization of 84

1. 84 / 2 = 42 (2 is a prime factor)
2. 42 / 2 = 21 (2 is a prime factor)
3. 21 / 3 = 7 (3 is a prime factor)
4. 7 / 7 = 1 (7 is a prime factor)

Therefore, the prime factorization of 84 is 2 x 2 x 3 x 7, which can also be written as 2² x 3 x 7.

Summary

Finding the prime factorization of a number involves systematically dividing the number by prime numbers until you reach 1. Start with the smallest prime number (2) and work your way up. Optimizations like only checking odd numbers and only checking up to the square root of the number can significantly improve the efficiency of this process.