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# Class Note for CSCI 124 with Professor Vora at GW (4)

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This 1 page Class Notes was uploaded by an elite notetaker on Saturday February 7, 2015. The Class Notes belongs to a course at George Washington University taught by a professor in Fall. Since its upload, it has received 19 views.

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Date Created: 02/07/15
CSCI 124 and CSCI 224VoIaGWU l CSCI 124224 Discrete Structures II Divisibility In this section we study the divisibility of one integer by another You should already be familiar with the basic ideas but need to be able to prove them using simple logical steps Again there is no way to become familiar with proofs without practice Make sure you do the problems from the handout 3 denotes there exists De nition For integers a and b a y 0 a is said to divide b if 3 integer m such that b ma This is denoted as all b is said to be divisible by a Also a is afactor or divisor of b which is a multiple of a Examples 2 1024 3 171 5 l 1024 5 does not divide 1024 Theorem Divisibility is transitive That is for integers a 127 e such that a y 0 and b y 0 if all and blc then ale Proof Suppose a y 0 and b y 0 Suppose further that all and blc all his 3 7721777227 sill mla c mgb 5 mlmga ale Example 9 126 and l26l378 hence 9 378 Division Theorem If a and b are integers such that b gt 0 3 unique integers q the quotient and r the remainder such that a bq r with 0 g r lt b We do not study its proof Example a 7 b 3 r 1 q 2 Another example a 79 b 5 q 72 r 1 andnotq 71 and r 74 Why not An even integer is an integer that is divisible by 2 That is there is an integer m such that the even integer may be written as 2m An odd integer is one that is not divisible by 2 Its remainder is 1 when divided by 2 hence 3 unique integer 4 such that the odd integer may be written as 24 1 If the product of two integers is the integer 1 both integers are either 1 or 71

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