Download 1 6-26-2013. Prime and composite numbers • A natural number a is

1 6‐26‐2013. Prime and composite numbers  A natural number a is divisible by a natural number b if_________________________________ o When a is divisible b we say that a is a ______________of b and b is a ______________ of a.  Example. 12 is divisible by 3 because ____________________  Example 2. Find all natural number factors 36.  A factorization of a natural number a is ______________________________ o Example. A factorization of 40 is 5  8. Another factorization of 40 is ________________  A prime number is a natural number greater than 1 whose only factors are ________________. o Example. The first five prime numbers are  A composite number has _____________________________________________________. o Example. The first five composite numbers are 2 
Sieve of Eratosthenes (Table 1) 11 21 31 41 51 61 71 81 91 2 3 4
5
6
7
8
9 10 12 13 14
15
16
17
18
19 20 22 23 24
25
26
27
28
29 30 32 33 34
35
36
37
38
39 40 42 43 44
45
46
47
48
49 50 52 53 54
55
56
57
58
59 60 62 63 64
65
66
67
68
69 70 72 73 74
75
76
77
78
79 80 82 83 84
85
86
87
88
89 90 92 93 94
95
96
97
98
99 100 
Divisibility tests (Table 2) Divisible by: Divisibility test 2 3 ________________________________ 4 ________________________________ 5 ________________________________ 6 8 ________________________________ ________________________________ ________________________________ 9 ________________________________ 10 ________________________________ Example 3 
The Fundamental Theorem of algebra: ____________________________________ _____________________________________________________________________ o
Example 5. Use a prime factor tree to find the prime factorization of 1320. 
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A mathematical proof is _____________________________________________________ o The fact that there are infinitely many prime numbers was proved by _____________ in approximately __________B.C. List of the first 100,008 primes: http://primes.utm.edu/lists/small/10000.txt Marin Mersenne (1588 – 1648) studied integers Mn = __________ and proved that: o If n is composite, then Mn __________________________  M4 = ____________________ o If n is prime, then Mn ______________________________  M2 = ____________ M3 =______________  M5 = ____________ M7 =____________ o If Mn is prime, we call it a _______________________ 4 o The Mersenne prime M11 =____________ was discovered in ________________ o By 1600 it was known that Mn is prime for n = ________________________ o The Mersenne prime M31 was discovered by ____________ around 1760 o The Mersenne prime M127 was discovered by ____________ in 1876 o In 1952 an early computer found Mersenne primes for n = 521, 607, 1279, 2203, 2281 o In 1996 the Great Internet Mersenne Prime Search (GIMPS) was launched. GIMPS currently involves about 50,000 PCs worldwide: http://primes.utm.edu/bios/page.php?id=42 o As of today, _________ Mersenne primes are known. o The largest known Mersenne prime is _______________ (17,425,170 digits) and was found on 2/05/2013