what is the prime power factorization
The prime power factorization, also known as the prime factorization or factorization into prime factors, is the process of expressing a positive integer as the product of its prime factors, each factor being raised to a power.
To find the prime power factorization of a number, you can follow these steps:
1. Start with the number you want to factorize.
2. Divide the number by the smallest prime number possible (2 is the smallest prime number).
3. If the number is divisible by 2, divide it by 2 and keep track of the number of times you divide it.
4. If the number is not divisible by 2, move on to the next prime number (3) and check if the number is divisible by it.
5. If the number is divisible by the current prime number, divide it and keep track of the number of times you divide it.
6. Continue this process, moving on to the next prime number each time, until the number becomes 1.
7. The prime power factorization of the original number is the product of the prime factors, each raised to the power corresponding to the number of times it was divided.
For example, let's find the prime power factorization of the number 36:
1. We start with 36.
2. 36 is divisible by 2, so we divide it by 2, resulting in 18.
3. 18 is still divisible by 2, so we divide it again, resulting in 9. We have divided by 2 twice so far.
4. 9 is not divisible by 2, so we move on to the next prime number, which is 3.
5. 9 is divisible by 3, so we divide it, resulting in 3. We have divided by 3 once so far.
6. 3 is now a prime number, so we stop dividing.
7. The prime power factorization of 36 is 2^2 * 3^2 (2 raised to the power 2 times and 3 raised to the power 2 times).
In general, the prime power factorization allows us to break down numbers into their fundamental building blocks, which are prime numbers. This is useful in various areas of mathematics, especially in topics like number theory and algebra.