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Factors & Multiples Copyright©2001 Lynda Greene Prime numbers & prime factorization

Prime Numbers

Before we learn to find the Prime Factors of a number, we need to know what a Prime Number is. A Prime Number is a number that can only be divided evenly (no remainders) by itself and the number 1. Examples: Let’s look at the factors of several numbers x 21 x 31 x 4 2 x 2 1 x 51 x 6 2 x 3 1 x 7 These two numbers (4 and 6) have more than one set of factors, so they are called “composite numbers” The other numbers (2, 3, 5, and 7) have only one set of factors each, the number one (1) and itself (2, 3, 5, or 7). These are Prime Numbers!

To find the Prime Factorization of any number, you must divide over and over by bigger and bigger prime numbers. But before you can do that you need to know which numbers are prime. Here is a list of the first 25 prime numbers. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 Memorizing tip #1: Look at the last digits in the middle two rows, they are almost exactly the same. (3 rd row: last number changed, 4 th row: has a number missing) Memorizing tip #2: 2 primes in the 20’s, 2 in the 30’s, 3 in the 40’s, (then it repeats) 2 in the 50’s, 2 in the 60’s, 3 in the 70’s, (starts to repeat again) 2 in the 80’s Grouping Pattern: two two three 20’s 30’s 40’s 50’s 60’s 70’s 80’s...pattern breaks down One way to do this is to have a list of prime numbers that you can refer to, but the easiest way (in the long run) is to memorize the prime numbers. (It helps to know your Prime Numbers later when you learn to reduce fractions, simplify square roots and factor polynomials)

Practice Problems: (Hit enter to see the answers) Label each number below as prime, composite or neither 1) 73 5) 51 2) 87 6) 23 3) 77 7) 2 4) 29 8) 1 Answers: 1) prime 2) composite 3) composite 4) prime 5) composite 6) prime 7) prime 8) neither

Prime Factorization

Prime Factorization: Breaking a number up into the smallest possible pieces. These pieces are called “prime factors” and they are a group of “prime numbers” that when multiplied together are equal to the original number. Example: The prime factorization for 72 is: 2 x 2 x 2 x 3 x 3 or (2 3 x 3 2 ) These expressions multiply together to give you 72 and 2 and 3 are both prime numbers.

Here is a list of the first 25 prime numbers. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 Example: Find the Prime Factorization for the number 36 Steps: 1)Divide 36 by 2 (the first prime number) 2)Divide the answer by 2 *Keep using 2 until it doesn’t divide evenly anymore* 3) Divide by 3, until it doesn’t divide evenly 4)Divide by 5, 7, 11,... (each of the prime numbers) STOP ! when you get a Prime Number on the bottom. 36 Note: Many teachers insist on using this upside-down division symbol for Prime Factorization, but it is still plain old division, just write the answer underneath instead of on top. 18 ) ) ) 2 3 Divide by 2 Divide by 3 Prime Number STOP! Can’t divide by 2 anymore, go to next Prime Number (3) The answer is made up of the prime numbers on the outside of the division symbols. They must be written with multiplication signs between them. ANSWER: 2 x 2 x 3 x 3 or 2 2 x 3 2 Check: 2 x 2 x 3 x 3 = 36 You will get the original number back if your answer is correct.

List of prime numbers. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 Example: Find the Prime Factorization for the number 42 Divide 42 by 2, then divide the answers by 2 until they won’t divide evenly anymore, then divide by the next prime number (3, 5, 7, 11,...). Stop when you get a Prime Number on the bottom ) 7 2 ) 3 Divide by 2 Divide by 3 ANSWER: 2 x 3 x 7 Check your answer, 2 x 3 x 7=42 Can’t divide by 2 anymore, go to next Prime Number (3) Prime Number STOP!

List of prime numbers. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 Example: Find the Prime Factorization for the number ) 21 2 ) 3 Divide by 2 Divide by 3 ANSWER: 2 x 3 x 3 x 7 or 2 x 3 2 x 7 Check your answer, 2 x 3 x 3 x 7 = 126 Can’t divide by 2 anymore, go to next Prime Number (3) Prime Number STOP! ) Divide by 3 3 7

List of prime numbers. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 Example: Find the Prime Factorization for the number ) Divide by 2 ANSWER: 2 x 2 x 5 x 11 Check your answer, 2 x 2 x 5 x 11 = 220 Can’t divide by 2 anymore, go to next Prime Number (3) Prime Number STOP! ) Divide by Can’t divide by 3 either, go to next Prime Number (5) 110 )

List of prime numbers. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 Example: Find the Prime Factorization for the number ) 3 7 Divide by 3 Divide by 7 ANSWER: 3 x 7 x 13 Check your answer, 3 x 7 x 13 = 273 Can’t divide by 2 at all, go to next Prime Number (3) Prime Number STOP! 13 Can’t divide by 3 anymore, go to next Prime Number (5) 91 ) Can’t divide by 5 either, go to next Prime Number (7)

Practice Problems: (Hit enter to see the answers) Find the prime factorization for the following numbers 1)105 2) 72 3)225 4) 135 5)90 6) 63 7) 154 8) 3234 Answers: 1) 3 x 5 x 7 2) 2 x 2 x 2 x 3 x 3 3) 3 x 3 x 5 x 5 4) 3 x 3 x 3 x 5 5) 2 x 3 x 3 x 5 6) 3 x 3 x 7 7) 2 x 7 x 11 8) 2 x 3 x 7 x 7 x 11

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