DISCRETE MATHEMATICS WITH APPLICATION (
DISCRETE MATHEMATICS WITH APPLICATION (
5th Edition
ISBN: 9780357097717
Author: EPP
Publisher: CENGAGE L
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Chapter 4.10, Problem 31ES

Exercises 28—32 refer to the following definition.

Definition: The least common multiple of two nonzero integers a and b, denotéd lcm(a, b), is the positive integer c such that

a. a | c and b | c

b. for all positive integers m, if a | m and b | m , then c ≤ m .

31. Prove that for all integers a and b, gcd ( a , b ) | lcm ( a , b ) .

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Chapter 4 Solutions

DISCRETE MATHEMATICS WITH APPLICATION (

Ch. 4.1 - An integer is even if, and only if,_______.Ch. 4.1 - An integer is odd if, and only if,____Ch. 4.1 - An integer n is prime if, and only if,_______Ch. 4.1 - The most common way to disprove a universal...Ch. 4.1 - Prob. 5TYCh. 4.1 - To use the method of direct proof to prove a...Ch. 4.1 - In 1-4 justify your answer by using the...Ch. 4.1 - In 1-4 justify your answer by using by the...Ch. 4.1 - In 1-4 justify your answers by using the...Ch. 4.1 - In 1-4 justify your answers by using the...
Ch. 4.1 - Prove the statements in 5-11. There are integers m...Ch. 4.1 - Prove the statements in 5-11. There are distinct...Ch. 4.1 - Prove the statements in 5—11. 7. There are real...Ch. 4.1 - Prob. 8ESCh. 4.1 - Prove the statements in 5-11. There is a real...Ch. 4.1 - Prob. 10ESCh. 4.1 - Prove the statements in 5-11. There is an integer...Ch. 4.1 - In 12-13, (a) write a negation for the given...Ch. 4.1 - In 12-13, (a) write a negation for the given...Ch. 4.1 - Prob. 14ESCh. 4.1 - Disprove each of the statements in 14-16 by giving...Ch. 4.1 - Disprove each of the statements in 14-16 by giving...Ch. 4.1 - In 17-20, determine whether the property is true...Ch. 4.1 - In 17-20, determine whether the property is true...Ch. 4.1 - In 17-20, determine whether the property is true...Ch. 4.1 - In 17-20, determine whether the property is true...Ch. 4.1 - Prob. 21ESCh. 4.1 - Prove the statement is 21 and 22 by the method of...Ch. 4.1 - Prob. 23ESCh. 4.1 - Each of the statements in 23—26 is true. For each....Ch. 4.1 - Prob. 25ESCh. 4.1 - Prob. 26ESCh. 4.1 - Fill in the blanks in the following proof....Ch. 4.1 - In each of 28-31: a. Rewrite the theorem in three...Ch. 4.1 - In each of 28-31: a. Rewrite the theorem in three...Ch. 4.1 - In each of 28-31: a. Rewrite the theorem in three...Ch. 4.1 - Theorem 4,1-2: The sum of any even integer and...Ch. 4.2 - The meaning of every variable used in a proof...Ch. 4.2 - Proofs should be written in sentences that are...Ch. 4.2 - Every assertion in a proof should be supported by...Ch. 4.2 - Prob. 4TYCh. 4.2 - A new thought or fact that does not follow as an...Ch. 4.2 - Prob. 6TYCh. 4.2 - Displaying equations and inequalities increases...Ch. 4.2 - Some proof-writing mistakes are...Ch. 4.2 - Prove the statements in 1-11. In each case use...Ch. 4.2 - Prove the statements in 1-11. In each case use...Ch. 4.2 - Prove the statements in 1-11. In each case use...Ch. 4.2 - Prob. 4ESCh. 4.2 - Prove the statements in 1-11. In each case use...Ch. 4.2 - Prove the statements in 1-11. In each case use...Ch. 4.2 - Prob. 7ESCh. 4.2 - Prove the statements in 1-11. In each case use...Ch. 4.2 - Prove the statements in 1-11. In each case use...Ch. 4.2 - Prob. 10ESCh. 4.2 - Prove the statements in 1-11. In each case use...Ch. 4.2 - Prove that the statements in 12—14 are false....Ch. 4.2 - Prove that the statements in 12—14 are false....Ch. 4.2 - Prove that the statements in 12-14 are false....Ch. 4.2 - Find the mistakes in the “proofs” shown in 15-19....Ch. 4.2 - Prob. 16ESCh. 4.2 - Prob. 17ESCh. 4.2 - Find the mistakes in the “proofs” show in 15-19....Ch. 4.2 - Find the mistakes in the “proofs” shown in 15-19....Ch. 4.2 - In 20-38 determine whether the statement is true...Ch. 4.2 - In 20-38 determine whether the statement is true...Ch. 4.2 - In 20-38 determine whether the statement is true...Ch. 4.2 - Prob. 23ESCh. 4.2 - Prob. 24ESCh. 4.2 - In 20-38 determine whether the statement is true...Ch. 4.2 - In 20-38 determine whether the statement is true...Ch. 4.2 - In 20-38 determine whether the statement is true...Ch. 4.2 - Prob. 28ESCh. 4.2 - Prob. 29ESCh. 4.2 - In 20-38 determine whether the statement is true...Ch. 4.2 - In 20-38 determine whether the statement is true...Ch. 4.2 - Prob. 32ESCh. 4.2 - Prob. 33ESCh. 4.2 - In 20-38 determine whether the statement is true...Ch. 4.2 - Prob. 35ESCh. 4.2 - Prob. 36ESCh. 4.2 - Prob. 37ESCh. 4.2 - Prob. 38ESCh. 4.2 - Suppose that integers m and n are perfect squares....Ch. 4.2 - Prob. 40ESCh. 4.2 - Prob. 41ESCh. 4.3 - To show that a real number is rational, we must...Ch. 4.3 - Prob. 2TYCh. 4.3 - Prob. 3TYCh. 4.3 - The numbers in 1—7 are all rational. Write each...Ch. 4.3 - The numbers in 1—7 are all rational. Write each...Ch. 4.3 - Prob. 3ESCh. 4.3 - The numbers in 1—7 are all rational. Write each...Ch. 4.3 - The numbers in 1—7 are all rational. Write each...Ch. 4.3 - The numbers in 1—7 are all rational. Write each...Ch. 4.3 - The numbers in 1—7 are all rational. Write each...Ch. 4.3 - The zero product property, says that if a product...Ch. 4.3 - Assume that a and b are both integers and that a0...Ch. 4.3 - Assume that m and n are both integers and that n0...Ch. 4.3 - Prove that every integer is a rational number.Ch. 4.3 - Prob. 12ESCh. 4.3 - Prob. 13ESCh. 4.3 - Consider the statement: The cube of any rational...Ch. 4.3 - Prob. 15ESCh. 4.3 - Determine which of the statements in 15—19 are...Ch. 4.3 - Prob. 17ESCh. 4.3 - Determine which of the statements in 15—19 are...Ch. 4.3 - Determine which of the statements in 15—19 are...Ch. 4.3 - Use the results of exercises 18 and 19 to prove...Ch. 4.3 - Prob. 21ESCh. 4.3 - Use the properties of even and odd integers that...Ch. 4.3 - Use the properties of even and odd integers that...Ch. 4.3 - Prob. 24ESCh. 4.3 - Derive the statements in 24-26 as corollaries of...Ch. 4.3 - Derive the statements in 24-26 as corollaries of...Ch. 4.3 - It is a fact that if n is any nonnegative integer,...Ch. 4.3 - Suppose a, b, c, and d are integers and ac ....Ch. 4.3 - Suppose a,b, and c are integers and x,y and z are...Ch. 4.3 - Prove that one solution for a quadratic equation...Ch. 4.3 - Prob. 31ESCh. 4.3 - Prove that for every real number c, if c is a root...Ch. 4.3 - Use the properties of even and odd integers that...Ch. 4.3 - Prob. 34ESCh. 4.3 - Prob. 35ESCh. 4.3 - In 35-39 find the mistakes in the “proofs” that...Ch. 4.3 - Prob. 37ESCh. 4.3 - In 35-39 find the mistakes in the "proofs” that...Ch. 4.3 - In 35-39 find the mistakes in the “proofs” that...Ch. 4.4 - TO show that a nonzero integer d divides an...Ch. 4.4 - To say that d divides n means the same as saying...Ch. 4.4 - Prob. 3TYCh. 4.4 - Prob. 4TYCh. 4.4 - Prob. 5TYCh. 4.4 - The transitivity of divisibility theorem says that...Ch. 4.4 - Prob. 7TYCh. 4.4 - Prob. 8TYCh. 4.4 - Prob. 1ESCh. 4.4 - Give a reason for your answer in each of 1-13,...Ch. 4.4 - Prob. 3ESCh. 4.4 - Give a reason for your answer in each of 1-13,...Ch. 4.4 - Give a reason for your answer in each of 1-13,...Ch. 4.4 - Prob. 6ESCh. 4.4 - Prob. 7ESCh. 4.4 - Prob. 8ESCh. 4.4 - Give a reason for your answer in each of 1-13,...Ch. 4.4 - Prob. 10ESCh. 4.4 - Prob. 11ESCh. 4.4 - Prob. 12ESCh. 4.4 - Give a reason for your answer in each of 1—13....Ch. 4.4 - Fill in the blanks in the following proof that for...Ch. 4.4 - Prove statements 15 and 16 directly from the the...Ch. 4.4 - Prob. 16ESCh. 4.4 - Prob. 17ESCh. 4.4 - Consider the following statement: The negative of...Ch. 4.4 - Show that the following statement is false: For...Ch. 4.4 - Prob. 20ESCh. 4.4 - For each statement in 20-32, determine whether the...Ch. 4.4 - Prob. 22ESCh. 4.4 - For each statement in 20-32, determine whether the...Ch. 4.4 - Prob. 24ESCh. 4.4 - For each statement in 20-32, determine whether the...Ch. 4.4 - Prob. 26ESCh. 4.4 - For each statement in 20-32, determine whether the...Ch. 4.4 - For each statement in 20-32, determine whether the...Ch. 4.4 - For each statements in 20-32, determine whether...Ch. 4.4 - For each statement in 20-32, determine whether the...Ch. 4.4 - For each statement in 20-32, determine whether the...Ch. 4.4 - For each statement in 20—32, determine whether the...Ch. 4.4 - Prob. 33ESCh. 4.4 - Consider a string consisting of a’s, b’s, and c’s...Ch. 4.4 - Two athletes run a circular track at a steady pace...Ch. 4.4 - It can be shown (see exercises 44-48) that an...Ch. 4.4 - Use the unique factorization theorem to write the...Ch. 4.4 - Let n=8,424. Write the prime factorization for n....Ch. 4.4 - Prob. 39ESCh. 4.4 - Prob. 40ESCh. 4.4 - How many zeros are at the end of 458.885 ? Explain...Ch. 4.4 - Prob. 42ESCh. 4.4 - At a certain university 2/3 of the mathematics...Ch. 4.4 - Prove that if n is any nonnegative integer whose...Ch. 4.4 - Prove that if n is any nonnegative nonnegative...Ch. 4.4 - Prob. 46ESCh. 4.4 - Prob. 47ESCh. 4.4 - Prove that for any nonnegative integer n, if the...Ch. 4.4 - Prob. 49ESCh. 4.4 - The integer 123,123 has the form abc, abc, where...Ch. 4.5 - The quotient-remainder theorem says that for all...Ch. 4.5 - Prob. 2TYCh. 4.5 - Prob. 3TYCh. 4.5 - Prob. 4TYCh. 4.5 - Prob. 5TYCh. 4.5 - Prob. 6TYCh. 4.5 - For each of the values of n and d given in 1-6,...Ch. 4.5 - For each of the values of n and d given in 1-6,...Ch. 4.5 - For each of the values of n and d given in 1-6,...Ch. 4.5 - For each of the values of n and d given in 1-6,...Ch. 4.5 - Prob. 5ESCh. 4.5 - For each of the values of n and d given in 1-6,...Ch. 4.5 - Evalute the expressions in 7-10 43div9 43mod9Ch. 4.5 - Evalute the expressions in7-10 50div7 50mod7Ch. 4.5 - Evalute the expressions in7-10 28div5 28mod5Ch. 4.5 - Prob. 10ESCh. 4.5 - Check the correctness of formula (4.5.1) given in...Ch. 4.5 - Justify formula (4.5.1) for general values of DayT...Ch. 4.5 - On a Monday a friend says he will meet you again...Ch. 4.5 - If today isTuesday, what day of the week will it...Ch. 4.5 - January 1,2000, was a Saturday, and 2000 was a...Ch. 4.5 - Prob. 16ESCh. 4.5 - Prove directky from the definitions that for every...Ch. 4.5 - Prove that the product of any two consecutive...Ch. 4.5 - Prove directly from the definitions that for all...Ch. 4.5 - Prob. 20ESCh. 4.5 - Suppose b is any integer. If bmod12=5 , what is...Ch. 4.5 - Suppose c is any integer. If c mod 15=3 , what is...Ch. 4.5 - Prove that for every integer n, if mod 5=3 then...Ch. 4.5 - Prove that for all integers m and n, if m mod 5=2...Ch. 4.5 - Prove that for all integrs a and b, if a mod 7=5...Ch. 4.5 - Prove that a necessary and sufficient and...Ch. 4.5 - Use the quotient-remainder theorem with divisor...Ch. 4.5 - Prove: Given any set of three consecutive...Ch. 4.5 - Use the quotient-remainder theorem with divisor...Ch. 4.5 - Use the quotient-remainder theorem with divisor...Ch. 4.5 - In 31-33, you may use the properties listed in...Ch. 4.5 - In 31-33, yoy may use the properties listed in...Ch. 4.5 - In 31-33, you may use the properties listed in...Ch. 4.5 - Given any integer n, if n3 , could n, n+2 , and...Ch. 4.5 - Prob. 35ESCh. 4.5 - Prove each of the statements in 35-43. The product...Ch. 4.5 - Prove each of the statements in 35-43. For any...Ch. 4.5 - Prove of the statements in 35-43. For every...Ch. 4.5 - Prove each of the statement in 35-43. Every prime...Ch. 4.5 - Prob. 40ESCh. 4.5 - Prob. 41ESCh. 4.5 - Prove each of the statements if 35-43. For all...Ch. 4.5 - Prob. 43ESCh. 4.5 - A matrix M has 3 rows and 4 columns. [ a 11 a 12 a...Ch. 4.5 - Prob. 45ESCh. 4.5 - Prob. 46ESCh. 4.5 - If m, n, and d are integers, d0 , and d(mn) , what...Ch. 4.5 - Prob. 48ESCh. 4.5 - Prob. 49ESCh. 4.5 - Prob. 50ESCh. 4.6 - Given any real number x, the floor of x is the...Ch. 4.6 - Prob. 2TYCh. 4.6 - Prob. 1ESCh. 4.6 - Compute x and x for each of the values of x in...Ch. 4.6 - Prob. 3ESCh. 4.6 - Compute x and x for each of the values of x in...Ch. 4.6 - Use the floor notation to express 259 div 11 and...Ch. 4.6 - If k is an integer, what is [k]? Why?Ch. 4.6 - If k is an integer, what is [k+12] ? Why?Ch. 4.6 - Prob. 8ESCh. 4.6 - Prob. 9ESCh. 4.6 - Prob. 10ESCh. 4.6 - Prob. 11ESCh. 4.6 - Prob. 12ESCh. 4.6 - Prob. 13ESCh. 4.6 - Prob. 14ESCh. 4.6 - Prob. 15ESCh. 4.6 - Some of the statements in 15-22 are true and some...Ch. 4.6 - Prob. 17ESCh. 4.6 - Prob. 18ESCh. 4.6 - Some of the statements is 15-22 are ture and some...Ch. 4.6 - Prob. 20ESCh. 4.6 - Prob. 21ESCh. 4.6 - Prob. 22ESCh. 4.6 - Prob. 23ESCh. 4.6 - Prob. 24ESCh. 4.6 - Prob. 25ESCh. 4.6 - Prob. 26ESCh. 4.6 - Prob. 27ESCh. 4.6 - Prob. 28ESCh. 4.6 - Prove each of the statements in 23-33. 29. For any...Ch. 4.6 - Prob. 30ESCh. 4.6 - Prob. 31ESCh. 4.6 - Prob. 32ESCh. 4.6 - Prob. 33ESCh. 4.7 - To prove a statement by contradiction, you suppose...Ch. 4.7 - Prob. 2TYCh. 4.7 - Prob. 3TYCh. 4.7 - Fill in the blanks in the following proof by...Ch. 4.7 - Is 10 an irrational numbre? Explain.Ch. 4.7 - Prob. 3ESCh. 4.7 - Use proof by contradiction to show that for every...Ch. 4.7 - Prob. 5ESCh. 4.7 - Prob. 6ESCh. 4.7 - Carefully formulate the negations of each of the...Ch. 4.7 - Fill in the blanks for the following proof that...Ch. 4.7 - a. When asked to prove that the difference of any...Ch. 4.7 - Let S be the statement: For all positive real...Ch. 4.7 - Let T be the statement: The sum of any two...Ch. 4.7 - Let R be the statement: The square root of any...Ch. 4.7 - Let S be the statement: The product of any...Ch. 4.7 - Let T be the statements: For every integer a, if...Ch. 4.7 - Do there exist integers a,b, and c such that a,b,...Ch. 4.7 - Prove each staement in 16-19 by contradiction. For...Ch. 4.7 - Prob. 17ESCh. 4.7 - Prove each statemtent in 16-19 by contradiction....Ch. 4.7 - Prove each statemet in 16-19 by contradiction. For...Ch. 4.7 - Fill in the blanks in the following proof by...Ch. 4.7 - Consider the statement “For everyinteger n, if n2...Ch. 4.7 - Consider the statement “For every real number r,...Ch. 4.7 - Prob. 23ESCh. 4.7 - Prove each of the statement in 23-24 in two ways:...Ch. 4.7 - Prob. 25ESCh. 4.7 - Use any method to prove the statements in 26-29....Ch. 4.7 - Use any method to prove the statements in 26-29....Ch. 4.7 - Use any method to prove the statements in 26-29....Ch. 4.7 - Prob. 29ESCh. 4.7 - Let n=53. Find an approximate value for n and...Ch. 4.7 - a. Prove by contraposition: For all positive...Ch. 4.7 - Prob. 32ESCh. 4.7 - The sieve of Eratosthenes, name after its...Ch. 4.7 - Prob. 34ESCh. 4.7 - Use proof by contradiction to show that every...Ch. 4.7 - Prob. 36ESCh. 4.8 - The ancient Greeks discovered that in a right...Ch. 4.8 - One way to prove that 2 is an irrational number is...Ch. 4.8 - One way to prove that there are infinitely many...Ch. 4.8 - Prob. 1ESCh. 4.8 - Prob. 2ESCh. 4.8 - Prob. 3ESCh. 4.8 - Prob. 4ESCh. 4.8 - Let S be the statement: The cube root of every...Ch. 4.8 - Prob. 6ESCh. 4.8 - Prob. 7ESCh. 4.8 - Prob. 8ESCh. 4.8 - Determine which statements in 6-16 are true and...Ch. 4.8 - Prob. 10ESCh. 4.8 - Determine which statements in 6-16 are true and...Ch. 4.8 - Determine which statements in 6-16 are true and...Ch. 4.8 - Determine which statements in 6-16 are true and...Ch. 4.8 - Prob. 14ESCh. 4.8 - Determine which statements in 6-16 are true and...Ch. 4.8 - Prob. 16ESCh. 4.8 - Prob. 17ESCh. 4.8 - a. Prove that for every integer a, if a3 is even...Ch. 4.8 - Use proof by contradiction to show that for any...Ch. 4.8 - Prob. 20ESCh. 4.8 - Prob. 21ESCh. 4.8 - Prove that 5 is irrational.Ch. 4.8 - Prob. 23ESCh. 4.8 - Prob. 24ESCh. 4.8 - Use the proof technique illustrated in exercise 24...Ch. 4.8 - Prob. 26ESCh. 4.8 - Prob. 27ESCh. 4.8 - Prob. 28ESCh. 4.8 - Suppose a is an integer and p is a prime number...Ch. 4.8 - Let p1,p2,p3,... be a list of all prime numbers in...Ch. 4.8 - Prob. 31ESCh. 4.8 - Prob. 32ESCh. 4.8 - Prove that if p1,p2...., and pn are distinct prime...Ch. 4.8 - Prob. 34ESCh. 4.8 - Prob. 35ESCh. 4.8 - Prob. 36ESCh. 4.8 - Prob. 37ESCh. 4.8 - Prob. 38ESCh. 4.9 - The toatl degree of a graph is defined as_____Ch. 4.9 - Prob. 2TYCh. 4.9 - In any graph the number of vertices of odd degree...Ch. 4.9 - Prob. 4TYCh. 4.9 - Prob. 5TYCh. 4.9 - Prob. 6TYCh. 4.9 - Prob. 1ESCh. 4.9 - Prob. 2ESCh. 4.9 - A graph has vertices of degrees 0,2,2,3, and 9....Ch. 4.9 - A graph has vertices of degrees ,1,1,4,4, and 6....Ch. 4.9 - In each of 5-13 either draw a graph with the...Ch. 4.9 - In each of 5-13 either draw a graph with the...Ch. 4.9 - In each of 5-13 either draw a graph with the...Ch. 4.9 - In each of 5-13 either draw a graph with the...Ch. 4.9 - In each of 5-13 either draw a graph with the...Ch. 4.9 - In each of 5-13 either draw a graph with the...Ch. 4.9 - In each of 5—13 either draw a graph with the...Ch. 4.9 - Prob. 12ESCh. 4.9 - Prob. 13ESCh. 4.9 - Prob. 14ESCh. 4.9 - A small social network contains three people who...Ch. 4.9 - a. In a group of 15 people, is it possible for...Ch. 4.9 - In a group of 25 people, is it possible for each...Ch. 4.9 - Is there a simple graph, each of whose vertices...Ch. 4.9 - Prob. 19ESCh. 4.9 - Draw K6, a complete graph on six vertices. Use the...Ch. 4.9 - In a simple graph, must every vertex have degree...Ch. 4.9 - Prob. 22ESCh. 4.9 - Recall that Km,n denotes a complete bipartite...Ch. 4.9 - A (general) bipartite graph G is a simple graph...Ch. 4.9 - Prob. 25ESCh. 4.10 - When an algorithm statement of the form x:=e is...Ch. 4.10 - Consider an algorithm statement of the following...Ch. 4.10 - Prob. 3TYCh. 4.10 - Prob. 4TYCh. 4.10 - Given a nonnegative integer a and a positive...Ch. 4.10 - Prob. 6TYCh. 4.10 - If r is a positive integer, then gcd (r,0)=_____Ch. 4.10 - Prob. 8TYCh. 4.10 - Prob. 9TYCh. 4.10 - Find the value of z when each of the algorithm...Ch. 4.10 - Prob. 2ESCh. 4.10 - Consider the following algorithm segment:...Ch. 4.10 - Prob. 4ESCh. 4.10 - Prob. 5ESCh. 4.10 - Prob. 6ESCh. 4.10 - Make a trace table to trace the action of...Ch. 4.10 - Prob. 8ESCh. 4.10 - Prob. 9ESCh. 4.10 - Prob. 10ESCh. 4.10 - Prob. 11ESCh. 4.10 - Prob. 12ESCh. 4.10 - Prob. 13ESCh. 4.10 - Use the Euclidean algorithm to hand-calculate the...Ch. 4.10 - Use the Euclidean algorithm to hand-calculate the...Ch. 4.10 - Use the Euclidean algorithm to hand-calculate the...Ch. 4.10 - Make a trace table to trace the action of...Ch. 4.10 - Make a trace table to trace the action of...Ch. 4.10 - Make a trace table to trace the action of...Ch. 4.10 - Prob. 20ESCh. 4.10 - Prob. 21ESCh. 4.10 - Prove that for all positive integers a and b, a|b...Ch. 4.10 - Prove that if a and b are integers, not both zero,...Ch. 4.10 - Prob. 24ESCh. 4.10 - Prob. 25ESCh. 4.10 - Prob. 26ESCh. 4.10 - An alternative to the Euclidean algorithm uses...Ch. 4.10 - Prob. 28ESCh. 4.10 - Prob. 29ESCh. 4.10 - Prob. 30ESCh. 4.10 - Exercises 28—32 refer to the following definition....Ch. 4.10 - Prob. 32ES
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