lect16_1 - Introduction to Number Theory Integers...

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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.
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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
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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.
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4 Divisors of 6 are: ± 1, ± 2, ± 3, ± 6. 0 2 d d 3 d 4 d - d - 3 d - 2 d Integers divisible by the positive integer d
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5 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 0 because a =1 a 4) If a | b and b> 0 and a >0, then a b 5) If m Z, m 0, then a | b if and only if ( m a )|( m b )
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6 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 .
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7 7) If a | b then a | b c . Proof . By the definition of divisibility a | b implies that there is some integer x such that b = a x .
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