A Problem Solving Approach To Mathematics For Elementary School Teachers (13th Edition)
13th Edition
ISBN: 9780135183885
Author: Rick Billstein, Shlomo Libeskind, Johnny Lott, Barbara Boschmans
Publisher: PEARSON
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Chapter 4.2A, Problem 10A
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Assume that a company is considering purchasing a machine for $50,000 that will have a five-year useful life and a $5,000 salvage value. The
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7. [10 marks]
Let G
=
(V,E) be a 3-connected graph. We prove that for every x, y, z Є V, there is a
cycle in G on which x, y, and z all lie.
(a) First prove that there are two internally disjoint xy-paths Po and P₁.
(b) If z is on either Po or P₁, then combining Po and P₁ produces a cycle on which
x, y, and z all lie. So assume that z is not on Po and not on P₁. Now prove that
there are three paths Qo, Q1, and Q2 such that:
⚫each Qi starts at z;
• each Qi ends at a vertex w; that is on Po or on P₁, where wo, w₁, and w₂ are
distinct;
the paths Qo, Q1, Q2 are disjoint from each other (except at the start vertex
2) and are disjoint from the paths Po and P₁ (except at the end vertices wo,
W1, and w₂).
(c) Use paths Po, P₁, Qo, Q1, and Q2 to prove that there is a cycle on which x, y, and
z all lie. (To do this, notice that two of the w; must be on the same Pj.)
Chapter 4 Solutions
A Problem Solving Approach To Mathematics For Elementary School Teachers (13th Edition)
Ch. 4.1 - Prob. 1MCCh. 4.1 - A customer wants to mail a package. The postal...Ch. 4.1 - Prob. 4MCCh. 4.1 - Prob. 5MCCh. 4.1 - A number in which each digit except 0 appears...Ch. 4.1 - Prob. 7MCCh. 4.1 - Three tiles marked x, yandz need three different...Ch. 4.1 - Prob. 10MCCh. 4.1 - Prob. 11MCCh. 4.1 - Jamila claims 10 is both even and odd because it...
Ch. 4.1 - Sabine claimed that a whole number is divisible by...Ch. 4.1 - Your seventh-grade class has just completed a unit...Ch. 4.1 - A student claims that a number is divisible by 24...Ch. 4.1 - Alejandro claims zero is neither even nor odd...Ch. 4.1 - Prob. 17MCCh. 4.1 - Andy has three cards, A, B, and C. Each card has...Ch. 4.1 - Prob. 2NAEPCh. 4.1A - If you multiply an even number by 2 and add 1, is...Ch. 4.1A - Classify each of the following statements as true...Ch. 4.1A - Use the divisibility test to test each of the...Ch. 4.1A - Using divisibility test, solve each of the...Ch. 4.1A - Find all possible single digits that can be placed...Ch. 4.1A - Find in the following blanks with the greatest...Ch. 4.1A - Prob. 7ACh. 4.1A - Without using Calculator, classify each of the...Ch. 4.1A - Classify each of the following statements as true...Ch. 4.1A - Find if possible a whole number solution that...Ch. 4.1A - Classify each of the following statements as true...Ch. 4.1A - An archery target consists of five concentric...Ch. 4.1A - To find the remainder when a number is divided by...Ch. 4.1A - If 45|n, what other whole numbers divide n? Why?Ch. 4.1A - Devise a test for divisibility by 16.Ch. 4.1A - A palindrome is a number that reads the same...Ch. 4.1A - Prove the test for divisibility by 9 for any...Ch. 4.1A - If the five-digit number aa248 is divisible by 9,...Ch. 4.1B - If you multiply an odd number by 2 and add 1, is...Ch. 4.1B - Without using a calculator, test each of the...Ch. 4.1B - Using divisibility test, solve each of the...Ch. 4.1B - Find all possible single digits that can be placed...Ch. 4.1B - Without using a calculator, classify each of the...Ch. 4.1B - Prob. 7ACh. 4.1B - Prob. 8ACh. 4.1B - Prob. 9ACh. 4.1B - Prob. 10ACh. 4.1B - The bookstore marked some notepads down from 2.00...Ch. 4.1B - Classify each of the following statements as true...Ch. 4.1B - In a football game, a touchdown with an extra...Ch. 4.1B - Classify each of the following statements as true...Ch. 4.1B - Compete the following table where n is the given...Ch. 4.1B - If 28|n, what other whole numbers divide n? Why?Ch. 4.1B - Devise a test for divisibility by 25.Ch. 4.1B - Use the casting out nines approach outlined in...Ch. 4.1B - Prob. 19ACh. 4.1B - Prove the test for divisibility by 9 for any...Ch. 4.1B - For what digit a is the five-digit number a6a14...Ch. 4.2 - Mathematical Connections Explain why the product...Ch. 4.2 - Mathematical Connections Explain why the product...Ch. 4.2 - Mathematical Connections In the Sieve of...Ch. 4.2 - Mathematical Connections Let M=2357+11131719,...Ch. 4.2 - Prob. 5MCCh. 4.2 - Prob. 6MCCh. 4.2 - Prob. 7MCCh. 4.2 - A perfect square is a number that can be written...Ch. 4.2 - Prob. 9MCCh. 4.2 - Prob. 10MCCh. 4.2 - Prob. 11MCCh. 4.2 - Mathematical Connections When students asked their...Ch. 4.2 - Mathematical Connections a. In which of the...Ch. 4.2 - To test for divisibility by 12, one student...Ch. 4.2 - Prob. 15MCCh. 4.2 - Prob. 16MCCh. 4.2 - Prob. 17MCCh. 4.2 - Prob. 18MCCh. 4.2 - Prob. 19MCCh. 4.2 - Prob. 20MCCh. 4.2 - Prob. 21MCCh. 4.2 - Prob. 22MCCh. 4.2 - Prob. 23MCCh. 4.2 - Prove that if a number is divisible by 45, then it...Ch. 4.2 - Prob. 1NAEPCh. 4.2 - Prob. 2NAEPCh. 4.2A - a. Fill in the missing numbers in the following...Ch. 4.2A - Prob. 2ACh. 4.2A - Find the prime factorization of the following...Ch. 4.2A - What is the greatest prime that must be considered...Ch. 4.2A - Question. Determine which of the following Number...Ch. 4.2A - Find the least natural number that is divisible by...Ch. 4.2A - Find the least three-digit whole number that has...Ch. 4.2A - Prob. 8ACh. 4.2A - Prob. 9ACh. 4.2A - Explain why 2332253 is not a prime factorization...Ch. 4.2A - Is it possible to find non-zero whole numbers...Ch. 4.2A - If 32n=26355473117, explain why 2357116 is a...Ch. 4.2A - Is 74113 a factor of 75113? Explain why or why...Ch. 4.2A - Prob. 14ACh. 4.2A - Prob. 15ACh. 4.2A - Briah and Emma are on the same swim team for the...Ch. 4.2A - A women with a basket of eggs finds that if she...Ch. 4.2A - A band of 17 pirates stole a sack of golden coins....Ch. 4.2A - A license plate is in two parts, a two-digit...Ch. 4.2A - The prime numbers 11 and 13 are twin primes...Ch. 4.2A - A whole number is a perfect number if the sum of...Ch. 4.2B - a. Fill in the missing numbers in the following...Ch. 4.2B - Use a factor tree to find the prime factorization...Ch. 4.2B - Find the prime factorization of the following...Ch. 4.2B - What is the greatest prime that must be considered...Ch. 4.2B - Determine which of the following numbers are...Ch. 4.2B - Find the least natural number that is divisible by...Ch. 4.2B - Find the greatest four-digit whole number that has...Ch. 4.2B - Prob. 8ACh. 4.2B - Find the prime factorization of each of the...Ch. 4.2B - Explain why 225392 is not a prime factorization...Ch. 4.2B - Is it possible to find non-zero whole numbers...Ch. 4.2B - If 2n=26355473117 explain why 2357116 is a factor...Ch. 4.2B - Prob. 13ACh. 4.2B - Prob. 14ACh. 4.2B - Mr. Wong wants to build a new rectangular patio....Ch. 4.2B - Prob. 16ACh. 4.2B - Jose has fewer than 100 baseball cards. If he puts...Ch. 4.2B - Isabelle, Noelle, and Juliette acquired a pile of...Ch. 4.2B - For a collection of red, green and blue beads,...Ch. 4.2B - A prime such as 7331 is a magic prime because any...Ch. 4.2B - A whole number is an abundant number if the sum of...Ch. 4.3 - Can two non-zero whole numbers have a greatest...Ch. 4.3 - Prob. 2MCCh. 4.3 - Prob. 3MCCh. 4.3 - Prob. 4MCCh. 4.3 - Is it true that every common divisor of two...Ch. 4.3 - Prob. 6MCCh. 4.3 - Prob. 7MCCh. 4.3 - Prob. 8MCCh. 4.3 - Prob. 9MCCh. 4.3 - Prob. 10MCCh. 4.3 - Explain why GCD(a,b) is always a divisor of...Ch. 4.3 - Prob. 12MCCh. 4.3 - Eleanor claims that the GCD(0,a)=0. Is she...Ch. 4.3 - A student claims the GCD(0,0) does not exist. Is...Ch. 4.3 - Prob. 15MCCh. 4.3 - Prob. 16MCCh. 4.3 - Prob. 17MCCh. 4.3 - Prob. 18MCCh. 4.3 - Prob. 19MCCh. 4.3 - Prob. 20MCCh. 4.3 - Prob. 21MCCh. 4.3 - Prob. 22MCCh. 4.3 - Prob. 1NAEPCh. 4.3A - Use colored rods to find the GCD and the LCM of 6...Ch. 4.3A - Prob. 2ACh. 4.3A - Find the GCD and the LCM for each of the following...Ch. 4.3A - Find the GCD for each pair of numbers using the...Ch. 4.3A - Find the LCM for each of the following groups of...Ch. 4.3A - The product of two numbers is 1734 and their GCD...Ch. 4.3A - The GCD of two numbers is 19 and their LCM is 228....Ch. 4.3A - Prob. 8ACh. 4.3A - Prob. 9ACh. 4.3A - To find GCD24, 20, 12, it is possible to find...Ch. 4.3A - Show that 97, 219, 988, 751 and 4 are relatively...Ch. 4.3A - Draw Venn diagrams to show the prime factors for...Ch. 4.3A - Find three pairs (a,b) such that LCM(a,b)=48.Ch. 4.3A - Find all whole numbers x such that GCD(49,x)=1 and...Ch. 4.3A - In Quinns dormitory room, there are three...Ch. 4.3A - A company is running a promotion on Facebook....Ch. 4.3A - If there were to be 9 boys and 6 girls at a party...Ch. 4.3A - Three motorcyclists ride around a circular course...Ch. 4.3A - Anna has made 42 chocolate chip cookies and 54...Ch. 4.3A - The front wheel of a tricycle has a circumference...Ch. 4.3A - Jazz has 72 yd of red ribbon and 42 yd of blue...Ch. 4.3B - Use colored rods to find the GCD and the LCM of 4...Ch. 4.3B - Find the GCD and LCM for each of the following...Ch. 4.3B - Find the GCD and LCM for each of the following...Ch. 4.3B - Find the GCD and LCM for each of the following...Ch. 4.3B - Find the LCM for each of the following groups of...Ch. 4.3B - The product of two numbers is 5880 and their GCD...Ch. 4.3B - The GCD of two numbers is 11 and their LCM is 330....Ch. 4.3B - Assume a and b are natural numbers and answer the...Ch. 4.3B - Classify each of the following statements as true...Ch. 4.3B - To find GCD 24, 20, 12, it is possible to find GCD...Ch. 4.3B - Show that 181, 345, 913 and 11 are relatively...Ch. 4.3B - Draw Venn diagrams to show the prime factors for...Ch. 4.3B - Find three pairs (a,b) such that LCM(a,b)=60.Ch. 4.3B - Find all whole numbers x such that GCD(25,x)=1 and...Ch. 4.3B - A light display has colored lights, each of which...Ch. 4.3B - A movie rental store gave a free popcorn to every...Ch. 4.3B - Bonita purchases 245 white bouncy balls, 238...Ch. 4.3B - A college runs two different bus routes starting...Ch. 4.3B - By selling cookies at 24c each, Cheng made enough...Ch. 4.3B - Gina runs every 4 days, cycles every 16 days, and...Ch. 4.3B - Kobe has collected 300 football card and 264...Ch. 4.CR - Classify each of the following statements as true...Ch. 4.CR - Test each of the following numbers for...Ch. 4.CR - Find the greatest digit if possible, that makes...Ch. 4.CR - Prob. 4CRCh. 4.CR - How can you tell whether a number is divisible by...Ch. 4.CR - Prob. 6CRCh. 4.CR - Find all the positive divisors of 144.Ch. 4.CR - Prob. 8CRCh. 4.CR - Prob. 9CRCh. 4.CR - Determine whether each of the following numbers is...Ch. 4.CR - Prob. 11CRCh. 4.CR - Prob. 12CRCh. 4.CR - Find the prime factorization of each of the...Ch. 4.CR - Prob. 14CRCh. 4.CR - Prob. 15CRCh. 4.CR - Find the GCD for each of the following pairs of...Ch. 4.CR - Find the LCM of each of the following gropus of...Ch. 4.CR - Prob. 19CRCh. 4.CR - Prob. 20CRCh. 4.CR - Prob. 21CRCh. 4.CR - Prob. 22CRCh. 4.CR - Two bells ring at 8:00A.M for the remainder of the...Ch. 4.CR - Midas has 120 gold coins and 144 silver coins. He...Ch. 4.CR - Jane and Ramon are running laps on a track. If...Ch. 4.CR - Prob. 26CRCh. 4.CR - Prob. 27CRCh. 4.CR - Prob. 28CRCh. 4 - NOW TRY THIS For any whole numbers aandb,...Ch. 4 - NOW TRY THIS If d(x+y+z) and dz, why d(x+y)?Ch. 4 - Prob. 3NTCh. 4 - NOW TRY THIS 4 Colored rods are used in the...Ch. 4 - Prob. 5NTCh. 4 - Prob. 6NT
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