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by: Cydney Conroy


Cydney Conroy

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This 1 page Study Guide was uploaded by Cydney Conroy on Thursday October 29, 2015. The Study Guide belongs to MATH 3000 at University of Colorado at Boulder taught by Staff in Fall. Since its upload, it has received 30 views. For similar materials see /class/231829/math-3000-university-of-colorado-at-boulder in Mathematics (M) at University of Colorado at Boulder.




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Date Created: 10/29/15
Introduction to Abstract Mathematics MATH 3000401 HANDOUT 3 October 24 2007 Review for Exam 2 Practice Problems 1 Prove that for arbitrary sets A and B B is a subset of A if and only if there exists a set X such that A 7 X B 2 Find the number of integers between 1 and 1000 that are divisible by a at least one of the numbers 12 45 50 b at least two of the numbers 12 45 50 c exactly one of the numbers 12 45 50 3 Prove the following statements by induction a 22 8h 7 5 4n2 7 n for all integers n 2 1 b For the Fibonacci sequence f1 f2 f see Exercise 7 on p 69 of the text y is a multiple of 3 for all integers h 2 1 c If 0 lt q lt i then 1 q lt1 2 q for all integers n 21 d If an integer a gt 1 has prime factorization a p111 10 p1 lt p2 lt lt pm are primes and r E N for all i 1 S i S m then the number of positive divisors of a is r1 1r2 1 rm 1 4 a Use the Euclidean Algorithm to nd integers st such that 415 25t 1 b Show that if s t are any other integers such that 415 25t 1 then 251575 and41 t7t 5 Prove that for arbitrary positive integers a and b a there exist integers in and n such that lcmab am and lcma b bn b m and n are uniquely determined and hcfmn 1 6 Prove that the following two conditions on an integer n gt 1 are equivalent 1 n is a prime power that is n pk for some prime p and some h E N ii for arbitrary integers a and b such that ab is a multiple of n either a is a multiple of n or some power b5 s E N of b is a multiple of n 7 TRUE or FALSE Justify your answer a For arbitrary sets A and B A 7 B U B A b There is no odd integer between 2 and 1010 that is the cube of an integer and also the 7th power of an integer c hcfab hcfa ba 7 b for all integers a and b d For arbitrary integers a and b lcma b lbl if and only if hcfa b lat


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