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# num01 - Number Theory The Division Algorithm The next...

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Number Theory The Division Algorithm The next result, the Division Algorithm, is a cornerstone of Number Theory. MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 15

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Number Theory The Division Algorithm The next result, the Division Algorithm, is a cornerstone of Number Theory. As an example, consider the statement 23 = 3 7 + 2. MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 15
Number Theory The Division Algorithm The next result, the Division Algorithm, is a cornerstone of Number Theory. As an example, consider the statement 23 = 3 7 + 2. One way of reading this is: when 23 is divided by 7, the quotient is 3 and the remainder is 2. MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 15

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Number Theory The Division Algorithm The next result, the Division Algorithm, is a cornerstone of Number Theory. As an example, consider the statement 23 = 3 7 + 2. One way of reading this is: when 23 is divided by 7, the quotient is 3 and the remainder is 2. Theorem ( The Division Algorithm ) Given integers a and b with b > 0 , there exist unique integers q and r such that a = qb + r 0 r < b q is called the quotient and r is called the remainder in the division of a by b. MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 15
Number Theory Proof of the Division Algorithm Consider the set S = f a xb : x is an integer and a xb 0 g . MAT 300 Awtrey MATHEMATICS AND STATISTICS 2 / 15

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Number Theory Proof of the Division Algorithm Consider the set S = f a xb : x is an integer and a xb 0 g . Since b > 0 is an integer, b 1. So the set S is nonempty since a ( j a j ) b a + j a j 0. MAT 300 Awtrey MATHEMATICS AND STATISTICS 2 / 15
Number Theory Proof of the Division Algorithm Consider the set S = f a xb : x is an integer and a xb 0 g . Since b > 0 is an integer, b 1. So the set S is nonempty since a ( j a j ) b a + j a j 0. Thus S is a nonempty subset of N . MAT 300 Awtrey MATHEMATICS AND STATISTICS 2 / 15

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Number Theory Proof of the Division Algorithm Consider the set S = f a xb : x is an integer and a xb 0 g . Since b > 0 is an integer, b 1. So the set S is nonempty since a ( j a j ) b a + j a j 0. Thus S is a nonempty subset of N . By the Well-Ordering Principle S has a least element, call it r 0. MAT 300 Awtrey MATHEMATICS AND STATISTICS 2 / 15
Number Theory Proof of the Division Algorithm Consider the set S = f a xb : x is an integer and a xb 0 g . Since b > 0 is an integer, b 1. So the set S is nonempty since a ( j a j ) b a + j a j 0. Thus S is a nonempty subset of N . By the Well-Ordering Principle S has a least element, call it r 0. By definition of S , this means there exists an integer q such that r = a qb 0. MAT 300 Awtrey MATHEMATICS AND STATISTICS 2 / 15

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Number Theory Proof of the Division Algorithm Consider the set S = f a xb : x is an integer and a xb 0 g . Since b > 0 is an integer, b 1. So the set S is nonempty since a ( j a j ) b a + j a j 0. Thus S is a nonempty subset of N . By the Well-Ordering Principle S has a least element, call it r 0. By definition of S , this means there exists an integer q such that r = a qb 0. We want to prove that r < b .
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