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Unformatted text preview: ECS20 Handout 8 January 25, 2011 1. If a and b are integers with a negationslash = 0, we say a divides b if there is an integer k such that b = ak . Notation: a | b a is called a factor of b and b is a multiple of a . We write a negationslash | b when a does not divide b . 2. Examples: 3 negationslash | 7, 3 | 12. 3. Theorem: Let a , b , c be integers, then if a | b and a | c , then a | ( b + c ) if a | b , then a | bc for all integers c if a | b and b | c , then a | c Proof: .... (do-it-yourself) 4. Theorem (The Division Algorithm): given integers a , d > 0, there is a unique q and r , such that a = d q + r, r < d. d is referred to as divisor, q is quotient and r is remainder. 5. Modular arithmetic: a mod d = r = the remainder after dividing a by d, where a and b are integers and d > 0. 6. Examples: 7 mod 3 = 1, since 7 = 3 2 + 1 3 mod 7 = 3, since 3 = 7 0 + 3 133 mod 9 = 2, since 133 = 9 ( 15) + 2. (Note: the remainder r = a mod d cannot be negative. Consequently, in this example, the remainder is notnegative....
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