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Primes, Composites and Tests for Divisibility Section 5.1 Prime numbers are important for many reasons, including cryptography, computer security and making division easier. Consider this division using prime factors rather than long division: 18,144 ÷ 2,592 = (2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 x 7) ÷ (2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3) = 7 To become an expert at this kind of division, you need to become an expert in primes and factoring. So what is a prime number? _____________________________________________________________. What does the Fundamental theorem of Arithmetic say: _______________________________________ _____________________________________________________________________________________ What else do we know about primes? (1) We can use the Sieve of _________________ to find the list of primes up to any number (but in elementary school we usually only draw the Sieve for primes to one-hundred. (pg 199) (2) The prime factor test guarantees that we only have to check for prime factors of n by checking the numbers less than or greater to _____. (pg 207) (3) The slickest proof ever that there is an infinite number of primes is found on (page 221). Euclid first wrote this proof down. Explain why it works: A number that is not prime has more than two factors. This number type is called _________________ . Is the number one prime? Is it composite? Is it neither? Is it both? Explain please: _________________ _____________________________________________________________________________________ Complete the Sieve of Eratosthenes here: 1 2 3 4 5 6 7 8 Complete the spiral, like Vi Hart does in this video. Mark off the primes and look for a pattern: 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 5 1 2 4 3 100 Your book gives one algorithm for factoring: the factor tree method, but there are other methods. Read about the factor tree on page 200. Using the number 60, as you see in the book, you could also factor by dividing “upside down”: 2| 60 divide 60 by any of its prime factors 5| 30 divide the remainder by any other prime factor that works 2| 15 continue dividing whenever the remainder is not a prime. 3| 5 because 5 is prime we stop. 60 = 2 x 5 x 2 x 3 x 5 What does a|b mean? (See page 201) _____________________________________________________ _____________________________________________________________________________________ Divisibility Tests To find all of these factors, you will want to know how and why the divisibility test in your book work. Write notes below: Why does the test for two work? __________________________________________________________ Why does the test for four work? _________________________________________________________ Why does the test for eight work? _________________________________________________________ Why does the test for three work? ________________________________________________________ Why does the test for nine work? _________________________________________________________ Why does the test for eleven work? _______________________________________________________ Why does the test for six work? ___________________________________________________________ What part of the theorem at the bottom of page 205 can be used to show that just because 4|12 and 6|12, we cannot then say that (4x6)|12? Homework due on Wednesday, January 9: 5.1B #1 5.1B #5 5.1B #10 5.1B #11 Use Excel to make this list easier to construct. Why does this work so well? DO NOT USE LONG DIVISION! Use the divisibility rules.