This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View
Full
Document
 Winter '08
 Staff

Click to edit the document details