92 Primes

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The building blocks of the natural numbers are the prime numbers A positive integer is called prime if it is not divisible by any integer except for 1 and itself The rst several primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, It is customary not to refer to 1 as a prime The rst prime number is 2, and 2 is the only even prime number It is an old theorem of Euclid that there are in fact in nitely many primes The fundamental theorem of arithmetic says that every natural number can be written as a product of primes (or powers of primes) in a unique way For example, 2520 = 23 32 5 7

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And there is no other way to factor 2520 into primes Notice that the factors 2 and 3 are repeated the factor 2 occurs three times and the factor 3 occurs twice The factors 5 and 7 occur once only

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93 Modular Arithmetic

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One of the most useful devices for studying the integers is modular arithmetic We say that two integers m and n are equivalent modulo k if the number m n is divisible by k This concept is best illustrated with some examples EXAMPLE 91 Let k = 2 The numbers 3 and 5 are equivalent modulo 2 just because 5 3 = 2 is divisible by 2 In the same way, 5 and 11 are equivalent modulo 2 because 11 5 = 6 is divisible by 2 We write 5 = 3 mod 2 and 11 = 5 mod 2 In fact it turns out that any two odd integers are equivalent modulo 2 because the difference of two odd integers will be an even number We also note that 4 and 12 are equivalent modulo 2 because 12 4 = 8 is divisible by 2 In the same way, 24 and 48 are equivalent modulo 2 because 48 24 = 24 is divisible by 2 In point of fact, any two even numbers are equivalent modulo 2 EXAMPLE 92 Let k = 12 The numbers 4 and 18 are not equivalent modulo 12 because 18 4 = 14 is not divisible by 12 However, the numbers 13 and 37 are equivalent modulo 12 because 37 13 = 24 is divisible by 12 The numbers 125 and 185 are equivalent modulo 12 because 185 125 = 60 is divisible by 12 The numbers 5 and 132 are not equivalent modulo 12 Modularity respects the arithmetic operations For example, if n=m+

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n mod k = m mod k + mod k Also, if n=m then n mod k = (m mod k) ( mod k) As an application of these last ideas, we can see quickly that 1347 does not divide 25168 For if 25168 = m 1347 then 25168 mod 3 = (m mod 3) (1347 mod 3) or 1 mod 3 = (m mod 3) 0 = 0 mod 3 This is impossible

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94 The Concept of a Group

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A group is a set G, or a collection of objects, together with a binary operation for combining them We usually denote the binary operation by We assume that this binary operation satis es certain basic and plausible properties: 1 Associativity If g, h, k G then g (h k) = (g h) k 2 Identity element There is a distinguished element e G such that, for all g G, e g = g e = g 3 Multiplicative inverse For each g G there is an element h G such that g h = h g = e Notice that we do not assume that a group is commutative; that is, we do not assume that g h = h g for all g, h G The property of associativity that we