1-Number Theory

1-Number Theory - # 0, c # 0. Prove or disprove: if ac|bc,...

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Number Theory 8/27/09
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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 3.4]. 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? 17 (c). 17 (b)....
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1-Number Theory - # 0, c # 0. Prove or disprove: if ac|bc,...

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