11fDivisionAlgorithm

11fDivisionAlgorithm - -1 so there are d possiblities for r...

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Division Algorithm Division Algorithm for N Remark. Think about taking an n N and dividing it by d N . What happens? Let’s look at an example: take n = 11 and divide it by d = 5 to get 11 5 = 2 1 5 or equivalently 11 5 = 2 + 1 5 or equivalently 11 = 5 × 2 + 1 . In general, one can divide n N by d N . Think of as: d q n . . . r to get n d = q r d or equivalently n d = q + r d or equivalently n = d q + r for some quotient q N ∪ { 0 } and some remainder r N ∪ { 0 } where 0 r < d . Theorem. Division Algorithm for N ( n N ) ( d N ) ( ! q N ∪ { 0 } ) ( ! r N ∪ { 0 } ) [ ( n = dq + r ) (0 r < d ) ] equivalently ( n N ) ( d N ) ( ! q N ∪ { 0 } ) ( ! r ∈ { 0 , 1 , . . . , d - 1 } ) [ n = dq + r ] Remark. Here: r ∈ { 0 , 1 , 2 , . . . , d
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Unformatted text preview: -1 } so there are d possiblities for r . Division Algorithm for Z Theorem. Division Algorithm for Z . (7 th edition, page 62) ( ∀ n ∈ Z ) ( ∀ d ∈ Z \ { } ) ( ∃ ! q ∈ Z ) ( ∃ ! r ∈ Z ) [ ( n = dq + r ) ∧ (0 ≤ r < | d | ) ] equivalently ( ∀ n ∈ Z ) ( ∀ d ∈ Z \ { } ) ( ∃ ! q ∈ Z ) ( ∃ ! r ∈ { , 1 , . . . , ( | d | -1) } ) [ n = dq + r ] Remark. Here: r ∈ { , 1 , 2 , . . . , ( | d | -1) } so there are d possiblities for r . 1...
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This note was uploaded on 12/13/2011 for the course MATH 300 taught by Professor Staff during the Fall '11 term at South Carolina.

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