handout8

handout8 - ECS20 Handout 8 1 If a and b are integers with a...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the 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: .... (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....
View Full Document

Page1 / 3

handout8 - ECS20 Handout 8 1 If a and b are integers with a...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online