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Unformatted text preview: 1 Introduction to Number Theory Integers: Z={3, 2, 1, 0, 1, 2, 3, } Operations: addition, multiplication, subtraction. Given any two integers a , b Z we can define a + b Z a  b Z a b Z Integers Z are closed under operations +,  , . Closure properties under +,  , If a , b are integers, then a + b , a  b , a b are integers. 2 Commutative Law If a , b are integers, then a + b = b + a ; a b = b a Associative Law If a , b are integers, then a +( b + c ) = ( a + b )+ c ; a ( b c ) = ( a b ) c Distributive Law If a , b are integers, then a ( b + c ) = a b+ a c Identity elements for addition and multiplication For all integer a , a +0= a ; a 1= a Additive inverse For all integer a , a + ( a ) = ( a )+ a = 0 3 Definition. Given two integers a , b Z, b 0, we say that a is divisible by b and denote it b  a , if there exists an integer n Z, such that a = b n . multiple of b divisor of a Example. If we divide 10 by 5, the result is an integer again. So, we say, that 10 is divisible by 5 because there exists an integer n (2 in this case) such that 10=5 n . But 10 is not divisible by 3 (within integer domain). If a , b Z, b 0, then it may be that ( a / b ) Z, so integers are not closed under division. 4 Some properties of divisibility 2) 1  a For any a Z : 1) a  0 For any a , b Z 3) If a  b and b  a , then a = b because 0= a because a =1 a 4) If a  b and b&gt; and a &gt;0, then a b 5) If m Z, m 0, then a  b if and only if ( m a )( m b ) 5 For any a , b , c 6) If a  b and b  c , then a  c. Proof. a  b b = a x for some x Z (by defn of divisibility) b  c c = b y for some y Z (by defn of divisibility) By substitution we have c = ( a x ) y. By associative law , c = a ( x y ). x y=k is an integer by the closure property of integers under multiplication. c = a k means that a  c . 6 7) If a  b then a  b c ....
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 Spring '09

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