CS2800-Number-theory_v.11

# CS2800-Number-theory_v.11 - Discrete Math CS 2800 Prof....

This preview shows pages 1–10. Sign up to view the full content.

1 Discrete Math CS 2800 Prof. Bart Selman [email protected] Module Number Theory Rosen, Sections 3-4 to 3-7.

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

View Full Document
2 The Integers and Division Of course, you already know what the integers are, and what division is… However: There are some specific notations, terminology, and theorems associated with these concepts which you may not know. These form the basics of number theory . Vital in many important algorithms today (hash functions, cryptography, digital signatures; in general, on-line security).
3 The divides operator New notation: 3 | 12 To specify when an integer evenly divides another integer Read as “ 3 divides 12 The not-divides operator: 5 | 12 To specify when an integer does not evenly divide another integer Read as “5 does not divide 12”

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

View Full Document
4 Divides, Factor, Multiple Let a , b Z with a 0 . Defn.: a | b a divides b : ( 5 c Z: b=ac ) “There is an integer c such that c times a equals b. Example: 3 | - 12 True , but 3 | 7 False . Iff a divides b , then we say a is a factor or a divisor of b , and b is a multiple of a . Ex.: “ b is even” :≡ 2| b . Is 0 even? Is −4?
5 Results on the divides operator If a | b and a | c, then a | (b+c) Example: if 5 | 25 and 5 | 30, then 5 | (25+30) If a | b, then a | bc for all integers c Example: if 5 | 25, then 5 | 25*c for all ints c If a | b and b | c, then a | c Example: if 5 | 25 and 25 | 100, then 5 | 100 (“common facts” but good to repeat for background)

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

View Full Document
6 Divides Relation Theorem: 2200 a , b , c Z : 1. a |0 2. ( a | b a | c ) a | ( b + c ) 3. a | b a | bc 4. ( a | b b | c ) a | c Corollary: If a, b, c are integers , such that a | b and a | c , then a | mb + nc whenever m and n are integers .
7 Proof of (2) Show 2200 a , b , c Z: ( a | b a | c ) a | ( b + c ) . Let a , b , c be any integers such that a | b and a | c , and show that a | ( b + c ) . By defn. of | , we know 5 s : b=as , and 5 t : c=at . Let s , t , be such integers. Then b + c = as + at = a ( s + t ) . So, 5 u : b + c = au , namely u = s + t . Thus a |( b + c ) . QED

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

View Full Document
Divides Relation Corollary: If a, b, c are integers , such that a | b and a | c , then a | mb + nc whenever m and n are integers . Proof: From previous theorem part 3 (i.e., a|b a|be) it follows that a | mb and a | nc ; again, from previous theorem part 2 (i.e., ( a | b a | c ) a | ( b + c )) it follows that a | mb + nc
The Division “Algorithm” Theorem: Division Algorithm --- Let a be an integer and d a positive integer. Then there are unique integers q and r , with 0 ≤r < d , such that a = dq+r . It’s really a theorem, not an algorithm… Only called an “algorithm” for historical reasons. q

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 01/16/2010 for the course CS 2800 at Cornell University (Engineering School).

### Page1 / 126

CS2800-Number-theory_v.11 - Discrete Math CS 2800 Prof....

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online