slides2 - CS 531 Fall 2007 CS531 Number Theory(That You...

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T 1 CS 531, Fall 2007 Copyright © William C. Cheng CS531 Number Theory (That You Already Know) Bill Cheng
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T 2 Number Theory CS 531, Fall 2007 Copyright © William C. Cheng let a, b Z , then a divides b (or equivalently, a is a divisor of b , or a is a factor of b ) if there exists c Z such that b = ac this is denoted by a|b The set of all integers Z = { ..., -3, -2, -1, 0, 1, 2, 3, ... } e.g., -3|18 , 173|0 let a, b Z , b 1 , division of a by b yields integer q (the quotient) and r (the remainder) such that a = qb + r 0 r < b q and r are unique r = a mod b = a - b a/b q = a div b = a/b c Z , c is a common divisor of a and b if c|a and c|b greatest common divisor is denoted by gcd(a,b) least common multiplier is denoted by lcm(a,b) lcm(a,b) = a b / gcd(a,b)
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T 3 Number Theory (Cont...) CS 531, Fall 2007 Copyright © William C. Cheng a, b Z , a and b are said to be relative prime (or coprime ) if gcd(a,b) = 1 p Z , p 2 , p is said to be prime if its only positive divisors are 1 and p otherwise, p is called composite there are infinite number of prime numbers
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