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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: .... (doityourself) 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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 Winter '08
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 Euclidean algorithm, 5min, 3min, 2min

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