1_NumberTheory - ≠ 0, c ≠ 0. Prove or disprove: if...

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Number Theory 8/25/11
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Division x is the smallest integer x. x is the largest integer x. 11.7 = 12; 11.7 = 11; -5.3 = -5; -5.3 = -6. Example: Does 9 divide 36? yes 9 | 36 True or False: 11 | 120? False Example: Let n and d be positive integers. How many positive integers not exceeding n are divisible by d?
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Theorem 1 . Let a, b, and c be integers and a 0. (i) If a|b and a|c, then a|(b+c). (ii) If a|b, then a|bc. (iii) If a|b and b|c, b 0, then a|c.
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Problem 8 [KR, Section 4.1]. Prove or disprove: if a|bc, then a|b or a|c. 7 . Let a, b, and c be integers, a
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Unformatted text preview: ≠ 0, c ≠ 0. Prove or disprove: if ac|bc, then a|b. Corollary 1 . Let a, b, and c be integers and a ≠ 0. If a|b and a|c, then a|(mb+nc) whenever m and n are integers. Division Algorithm Let a and d be integers with d ≠ 0. Then there exist unique integers q and r, 0 ≤ r < |d|, such that a= q.d+r. Terminology a = dividend, d = divisor, q = quotient, and r = remainder q = a div d & r = a mod d Problem 9 b . What are the quotient and remainder when -111 is divided by 11? 21 (c). 21 (b)....
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This note was uploaded on 03/21/2012 for the course CS 3333 taught by Professor Boppana during the Fall '11 term at The University of Texas at San Antonio- San Antonio.

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1_NumberTheory - ≠ 0, c ≠ 0. Prove or disprove: if...

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