Prime Factorization - Secrets Revealed in Number Theory

Have you ever thought what other uses are there in knowing the prime factorization (PF) of natural numbers? Your teacher probably told you to find the prime factorization of certain numbers, but you never really thought of why do you need this for? First, let me define what are prime numbers? Prime numbers are numbers greater than one that has exactly two factors, one and the number itself. Numbers that have more than two factors are known as composite numbers. The smallest composite number is 4. The factors of 4 are 1, 2 and 4. Since 4 has three factors, 4 is considered a composite number. Now, did you know that all composite numbers can be expressed as the product of prime numbers? You probably did not even think of that? But this is what prime factorization is for? But I am not here to explain how to prime factorize numbers. I assume you already know how to do it. What I am going to do is to show you more about this which your school teacher probably never told you about? Unfortunately, some of them never even really thought about it? Secrets Revealed in Number Theory Prime factorization is one of the many lessons in Number Theory. In this lesson I am going to reveal two secrets about this. Secret #1 - Determining the Number of Factors of a Number To determine the number of factors of a number, first find the PF (include the 1 if a factor is to the first power). Now, increase each exponent (or index) by 1 and multiply these new numbers together. Example: How many factors does 480 have? The PF of 480 is 2^5 x 3^1 x 5^1. Increase each of the exponents (or indices) by 1 and multiply these new values: (5 + 1) x (1 + 1) x (1 + 1) = 6 x 2 x 2 = 24 So, 480 has 24 factors. (They are: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 80, 96, 120, 160, 240 and 480) Secret #2 - Finding the Sum of the Factors of a Number Example: What is the sum of the factors of 4200? From the PF 2^3 x 3^1 x 5^2 x 7^1, we know that 4200 has 4 x 2 x 3 x 2 = 48 factors. The sum of these 48 factors can be calculated from the PF, too: (2^0 + 2^1 + 2^2 + 2^3) x (3^0 + 3^1) x (5^0 + 5^1 + 5^2) x (7^0 + 7^1) = (1 + 2 + 4 + 8) x (1 + 3) x (1 + 5 + 25) x (1 + 7) = 15 x 4 x 31 x 8 =14 880 Now you know there are interesting things about prime factorization.