4383_Notes_2o2

4383_Notes_2o2 - a qb r = + with b < Examples:...

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Math 4383 Section 2.2 Page 1 of 3 Number Theory Chapter 2 Section 2.2: The Division Algorithm Remember that the basic premise of number theory is the study of the properties of the integers. So, when we perform arithmetic operations, we are looking for integer answers. We can add, subtract and multiply and always get an integer answer. However, we cannot divide two integers and always get an integer answer. So, we are going back to the basics of division. 12 divided by 3 means – What about 12 divided by 5 Division with quotient and remainder The Division Algorithm Given integers a and b with b>0, there exists unique integers q and r such that a qb r = + and 0 b < Terminology- Proof – It is in the book, I am not going to prove it in class because I prove this in Math 3330, and so many of you have already seen it (or will in about 2 weeks).
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Math 4383 Section 2.2 Page 2 of 3 The more general form: Given integers a and b with b nonzero, there exists unique integers q and r such that
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Unformatted text preview: a qb r = + with b &lt; Examples: Why is the division algorithm useful? Show that the square of any integer has the form 4k or 4k+1. Note that if we want to prove something about the divisibility of integers, we cant test all of them. But, if an integer is divided by n how many possible remainders are there? Math 4383 Section 2.2 Page 3 of 3 In the case of n = 7 EVERY INTEGER is of EXACTLY ONE of the following forms: If I want to prove that the cube of every integer is of the form 7k or 7k+1 Note that if we define a relation on the set of integers defined by x is related to y if they have the same remainder when divided by n this is an equivalence relation. The distinct equivalence classes are indexed by the remainders 0 n-1. EVERY integer falls into EXACTLY ONE of these equivalence classes (the relation PARTITIONS the set of integers)....
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4383_Notes_2o2 - a qb r = + with b &amp;amp;lt; Examples:...

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