# Prime Factorization

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## Prime Factorization

Exploring Integers - Part 6

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 Properties of Integers: I.  Prime Factorization     a. divisors         'd'  is a divisor of 'a' if there is another integer c  such that          c * d = a      Example:     5 is a divisor of 15 since 3 * 5  = 15     Example:     3 is a divisor of 15 since  5 * 3 = 15     Example:     2 is not a divisor of 15 since there is no number to                        multiply 2 by to equal 15    b. prime number         if p is an number other than 0, 1, or -1, and it has no divisors         other than 1 or -1, then p is called a prime number     Example:  20   is not a prime number since 2 and 5 are divisors of 20     Example:  3, 5, 7, 11, 13, 17, 19  are prime numbers       FUNDAMENTAL THEOREM OF ARITHMETIC:     Any positive integer greater than one can be expressed as a product of positive prime integers in only one way (not including order). note:  these prime numbers are called factors note:  the product can be a single number, the prime number itself. Example: 12  = 2 * 6,  6 is not prime, so we have  2 * (2 * 3) = 2*2*3  Example: 13  =  13 Example: 42  = 2 * 21  = 2 * (3 * 7) = 2*3*7 Example: 81  = 3 * 27  = 3 * (3 * 9) = 3*3*(3 * 3) = 3*3*3*3    c.  Exponent It will become cumbersome to write repeated factors of larger numbers, so for notational convenience, if a prime is repeated, we can write it as that prime with a superscript that is the number of times that prime is repeated. Example:  3 * 3 * 3 * 3  will be written as  34 Example:  2 * 2  =  22 Example:  13  =  131   (no superscript implies a superscript of 1) Example: 648 = 2 * 324 = 2 * 2 * 162                      = 2 * 2 * 2 * 81 = 2*2*2*3*3*3*3  = 23*34 So, 23*34  MEANS  2*2*2*3*3*3*3 Hints for finding the prime factorization: 1.  Start with the prime divisors 2, 3, 5, 7,  11, 13, etc, in order, until you reach the square root of the number (more on this later) 2.  Recognize that if 2 divides the last digit in the number, 2 is a factor 3.  #2 is also true for 5 4.  If you add up the digits in a number and 3 divides that sum then 3 divides that number. Example:     12   (last digit is 2, 2 divides 2 so start with 2)                    12 = 2 * 6   (look at 6, 2 divides 6 so we have)                    12 = 2 * ( 2 * 3 )     ( look at 3, 3 is prime, we're done)                    12 = 22 * 3  Example:      30           ( 2 divides 0 )                     30 = 2 * 15       ( look at 15,  1 + 5 = 6 and 3 divides 6)                     30 = 2 * ( 3 * 5 )      (look at 5, 5 is prime, we're done )                     30 = 2 * 3 * 5  Index Back Continue Last page   This is another FREE ALGEBRA PRINTABLE presented to you from the Algebra section of K12math.com