Download Primes, Composites and Tests for Divisibility

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.