What is the prime power factorization.
The prime power factorization is a way to express a positive integer as a product of prime numbers raised to certain powers. To find the prime power factorization of a number, follow these steps:
1. Start with the given number that you want to factorize.
2. Begin dividing the number by the smallest prime number, which is 2. If the number is divisible by 2, divide it as many times as possible until it is no longer divisible.
3. Move on to the next prime number, which is 3. Repeat the process of dividing the number until it is no longer divisible by 3.
4. Continue this process with each prime number, incrementing by one each time (i.e., 5, 7, 11, and so on) until you have divided the number as much as possible.
5. Keep track of the prime factors you have found, along with their corresponding powers.
6. Once you cannot divide the number any further, the prime power factorization is complete. Write the number as a product of the prime factors, each raised to its respective power.
For example, let's find the prime power factorization of the number 180:
1. Start with 180.
2. Divide by 2: 180 ÷ 2 = 90. Since 90 is still divisible by 2, repeat: 90 ÷ 2 = 45.
3. Divide by 3: 45 ÷ 3 = 15. Now 15 is not divisible by 2 or 3.
4. Divide by 5: 15 ÷ 5 = 3. Finally, 3 is a prime number, so we stop here.
5. The prime factors we found are 2, 2, 3, 3, and 5.
6. The prime power factorization of 180 is 2² × 3² × 5, or 2^2 × 3^2 × 5.
Note that the prime power factorization is unique for each positive integer, meaning that every positive integer can be expressed as a product of prime numbers raised to specific powers.