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cours - matière potentielle : students
cours - matière potentielle : application
cours - matière potentielle : contexts
cours - matière potentielle : programs
cours - matière potentielle : environments
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Yusra L Visser Page 1 4/5/2002 Effects of Problem-Based and Lecture-Based Instructional Strategies on Problem Solving Performance and Learner Attitudes in a High School Genetics Class

learning as an instructional strategy
primary concerns with the potential generalizability of the findings on problem
performance
activities
learners
findings
research
skills
problem
students

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Modular Arithmetic

Normally we think of arithmetic using the entire (innite) set of integers. But when using computers to perform arithmetic we must settle for using only a nite set of integers. For example, under normal cicumstances, numbers are stored as 32 or 64-bit words in computer memory. Thus we must be careful to ensure that the basic arithmetical operations (namely, addition, subtraction, and multiplication) remain closed under the nite set of integers that is used. In other words, ifS⊂Iis a nite set of integers, andx, y∈S, thenx+y, xy, x− y∈Sthese operations will not be well-dened with respect to. Otherwise, S.Modular arithmetic, a fundamental subject in number theory and other areas of higher mathematics, gives us precisely the framework we need to ensure closure of operations while working with a nite set. Moreover, we will witness several applications of modular arithmetic to computer science and engineering.

Letaandbbe integers andm >1 a positive integer. We say thata≡bmodmi one of the following conditions holds

1.m|a−b

2. there is some integerksuch thata=mk+b

3.amdom=bdomm

Equivalently we say thataiscongruenttobmodm. Integermis often called themodulus.

Theorem 1.Conditions 1,2, and 3 are all equivalent. In other words, for givena,b, andm, they are either all true, or all false.

Proof of Theorem 1.

Exam

1.Is it true that 3≡Give at least 5 numbers that are con15 mod 12?

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In arithmetic with rational numbers amluit

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Interestin facts twos-com

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ve these!

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