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College Algebra Exam Review 19

College Algebra Exam Review 19 - 1.6 DIVISIBILITY IN THE...

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1.6. DIVISIBILITY IN THE INTEGERS 29 existence part of the following assertion. In the following proof, I’m go- ing to arrange the inductive argument in a more streamlined way, but the essential idea is the same. Proposition 1.6.7. Given integers a and d , with d 1 , there exist unique integers q and r such a D qd C r and 0 r < d . Proof. First consider the case a 0 . If a < d , take q D 0 and r D a . Now suppose that a d . Assume inductively that the existence assertion holds for all nonnegative integers that are strictly smaller than a . Then in particular it holds for a d , so there exist integers q 0 and r such that .a d/ D q 0 d C r and 0 r < d . Then a D .q 0 C 1/d C r , and we are done. We deal with the case a < 0 by induction on j a j . If d < a < 0 , take q D 1 and r D a C d . Suppose that a d , and assume inductively that the existence assertion holds for all nonpositive integers whose absolute values are strictly smaller than j a j . Then in particular it holds for a C d , so there exist integers q 0 and r such that .a C d/ D q 0 d C r and 0 r < d .
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