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CS2800-Number-theory_v.11

CS2800-Number-theory_v.11 - Discrete Math CS 2800 Prof Bart...

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1 Discrete Math CS 2800 Prof. Bart Selman [email protected] Module Number Theory Rosen, Sections 3-4 to 3-7.
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2 The Integers and Division Of course, you already know what the integers are, and what division is… However: There are some specific notations, terminology, and theorems associated with these concepts which you may not know. These form the basics of number theory . Vital in many important algorithms today (hash functions, cryptography, digital signatures; in general, on-line security).
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3 The divides operator New notation: 3 | 12 To specify when an integer evenly divides another integer Read as “ 3 divides 12 The not-divides operator: 5 | 12 To specify when an integer does not evenly divide another integer Read as “5 does not divide 12”
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4 Divides, Factor, Multiple Let a , b Z with a 0 . Defn.: a | b a divides b : ( 5 c Z: b=ac ) “There is an integer c such that c times a equals b. Example: 3 | - 12 True , but 3 | 7 False . Iff a divides b , then we say a is a factor or a divisor of b , and b is a multiple of a . Ex.: “ b is even” :≡ 2| b . Is 0 even? Is −4?
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5 Results on the divides operator If a | b and a | c, then a | (b+c) Example: if 5 | 25 and 5 | 30, then 5 | (25+30) If a | b, then a | bc for all integers c Example: if 5 | 25, then 5 | 25*c for all ints c If a | b and b | c, then a | c Example: if 5 | 25 and 25 | 100, then 5 | 100 (“common facts” but good to repeat for background)
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6 Divides Relation Theorem: 2200 a , b , c Z : 1. a |0 2. ( a | b a | c ) a | ( b + c ) 3. a | b a | bc 4. ( a | b b | c ) a | c Corollary: If a, b, c are integers , such that a | b and a | c , then a | mb + nc whenever m and n are integers .
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7 Proof of (2) Show 2200 a , b , c Z: ( a | b a | c ) a | ( b + c ) . Let a , b , c be any integers such that a | b and a | c , and show that a | ( b + c ) . By defn. of | , we know 5 s : b=as , and 5 t : c=at . Let s , t , be such integers. Then b + c = as + at = a ( s + t ) . So, 5 u : b + c = au , namely u = s + t . Thus a |( b + c ) . QED
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Divides Relation Corollary: If a, b, c are integers , such that a | b and a | c , then a | mb + nc whenever m and n are integers . Proof: From previous theorem part 3 (i.e., a|b a|be) it follows that a | mb and a | nc ; again, from previous theorem part 2 (i.e., ( a | b a | c ) a | ( b + c )) it follows that a | mb + nc
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9 The Division “Algorithm” Theorem: Division Algorithm --- Let a be an integer and d a positive integer. Then there are unique integers q and r , with 0 ≤r < d , such that a = dq+r . It’s really a theorem, not an algorithm… Only called an “algorithm” for historical reasons. q is called the quotient r is called the remainder d is called the divisor a is called the dividend
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10 What are the quotient and remainder when 101 is divided by 11? q is called the quotient r is called the remainder d is called the divisor a is called the dividend 101 = 11 × 9 + 2 We write: q = 9 = 101 div 11 r = 2 = 101 mod 11 a d q r
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11 If a = 7 and d = 3, then q = 2 and r = 1, since 7 = (2)(3) + 1.
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