Mathematical Ideas (13th Edition) - Standalone book
13th Edition
ISBN: 9780321977076
Author: Charles D. Miller, Vern E. Heeren, John Hornsby, Christopher Heeren
Publisher: PEARSON
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Chapter 5.3, Problem 33E
To determine
The one pair of twin primes between
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b) Solve the following linear program using the 2-phase simplex algorithm. You should give
the initial tableau, and each further tableau produced during the execution of the
algorithm. If the program has an optimal solution, give this solution and state its
objective value. If it does not have an optimal solution, say why.
maximize ₁ - 2x2+x34x4
subject to 2x1+x22x3x41,
5x1 + x2-x3-×4 ≤ −1,
2x1+x2-x3-34
2,
1, 2, 3, 40.
Suppose we have a linear program in standard equation form
maximize cTx
subject to Ax = b.
x ≥ 0.
and suppose u, v, and w are all optimal solutions to this linear program.
(a) Prove that zu+v+w is an optimal solution.
(b) If you try to adapt your proof from part (a) to prove that that u+v+w
is an optimal solution, say exactly which part(s) of the proof go wrong.
(c) If you try to adapt your proof from part (a) to prove that u+v-w is an
optimal solution, say exactly which part(s) of the proof go wrong.
a) Suppose that we are carrying out the 1-phase simplex algorithm on a linear program in
standard inequality form (with 3 variables and 4 constraints) and suppose that we have
reached a point where we have obtained the following tableau. Apply one more pivot
operation, indicating the highlighted row and column and the row operations you carry
out. What can you conclude from your updated tableau?
x1
x2 x3
81 82
83
84
81
-2 0
1 1 0
0
0
3
82
3 0
-2 0
1
2
0
6
12
1
1
-3
0
0
1
0
2
84
-3 0
2
0
0 -1
1
4
-2 -2 0
11
0
0-4
0
-8
Chapter 5 Solutions
Mathematical Ideas (13th Edition) - Standalone book
Ch. 5.1 - Decide whether each statement is true or false
1....Ch. 5.1 - Decide whether each statement is true or false. If...Ch. 5.1 - Decide whether each statement is true or false....Ch. 5.1 - Prob. 4ECh. 5.1 - Prob. 5ECh. 5.1 - Prob. 6ECh. 5.1 - Decide whether each statement is true or false.
7....Ch. 5.1 - Prob. 8ECh. 5.1 - Prob. 9ECh. 5.1 - Find all natural number factors of each...
Ch. 5.1 - Find all natural number factors of each number. 28Ch. 5.1 - Find all natural number factors of each number. 72Ch. 5.1 - Use divisibility tests to decide whether the given...Ch. 5.1 - Use divisibility tests to decide whether the given...Ch. 5.1 - Use divisibility tests to decide whether the given...Ch. 5.1 - Prob. 16ECh. 5.1 - (a) In constructing the Sieve of Eratosthenes for...Ch. 5.1 - (a) Continue the Sieve of Eratosthenes in Table 1...Ch. 5.1 - In your list for Exercise 18(a). consider the six...Ch. 5.1 - Prob. 20ECh. 5.1 - Prob. 21ECh. 5.1 - Prob. 22ECh. 5.1 - Prob. 23ECh. 5.1 - Prob. 24ECh. 5.1 - Prob. 25ECh. 5.1 - Prob. 26ECh. 5.1 - Prob. 27ECh. 5.1 - Find the prime factorization of each composite...Ch. 5.1 - Prob. 29ECh. 5.1 - Prob. 30ECh. 5.1 - Here is a divisibility test for 7.
(a) Double the...Ch. 5.1 - Here is a divisibility test for 7. (a)Double the...Ch. 5.1 - Prob. 33ECh. 5.1 - Prob. 34ECh. 5.1 - Here is a divisibility test for 11. (a) Starting...Ch. 5.1 - Prob. 36ECh. 5.1 - Here is a divisibility test for 11.
(a) Starting...Ch. 5.1 - Prob. 38ECh. 5.1 - 39. Consider the divisibility test for the...Ch. 5.1 - 40. Give two factorizations of the number 75 that...Ch. 5.1 - Prob. 41ECh. 5.1 - Determine all possible digit replacements for x so...Ch. 5.1 - Determine all possible digit replacements for x so...Ch. 5.1 - Determine all possible digit replacements for x so...Ch. 5.1 - Prob. 45ECh. 5.1 - Prob. 46ECh. 5.1 - Prob. 47ECh. 5.1 - Prob. 48ECh. 5.1 - Prob. 49ECh. 5.1 - Prob. 50ECh. 5.1 - Leap years occur when the year number is divisible...Ch. 5.1 - Prob. 52ECh. 5.1 - Prob. 53ECh. 5.1 - Leap years occur when the year number is divisible...Ch. 5.1 - Prob. 55ECh. 5.1 - Prob. 56ECh. 5.1 - Prob. 57ECh. 5.1 - 58. Choose any 6-digit number consisting of three...Ch. 5.1 - One of the authors has three sons who were born....Ch. 5.1 -
Ore of the authors has three sons who were born,...Ch. 5.1 - Prob. 61ECh. 5.1 - Prob. 62ECh. 5.2 - In Exercises 1-6 decide whether each statement is...Ch. 5.2 - In Exercises 1-6 decide whether each statement is...Ch. 5.2 - In Exercises 1-6 decide whether each statement is...Ch. 5.2 - Prob. 4ECh. 5.2 - In Exercises 1-6 decide whether each statement is...Ch. 5.2 - Prob. 6ECh. 5.2 - Prob. 7ECh. 5.2 - Prob. 8ECh. 5.2 - Prob. 9ECh. 5.2 - Prob. 10ECh. 5.2 - Prob. 11ECh. 5.2 - Prob. 12ECh. 5.2 - Prob. 13ECh. 5.2 - Prob. 14ECh. 5.2 - 15. (a) Evaluate the Fermat number for .
(b) In...Ch. 5.2 - 16. (a) Verify the value given in the text for the...Ch. 5.2 - Prob. 17ECh. 5.2 - Prob. 18ECh. 5.2 - 19. Why do you suppose it normally takes up to a...Ch. 5.2 - Prob. 20ECh. 5.2 - Prob. 21ECh. 5.2 - 22. Explain n your own words the proof by Euclid...Ch. 5.2 - 23. For the composite number , find
Ch. 5.2 - Prob. 24ECh. 5.2 - Prob. 25ECh. 5.2 - Prob. 26ECh. 5.2 - Prob. 27ECh. 5.2 - Prob. 28ECh. 5.2 - Explain why large prime numbers are important in...Ch. 5.2 - 30. Describe the difference between Mersenne...Ch. 5.3 - In Exercises 1-10 decide whether each statement is...Ch. 5.3 - Prob. 2ECh. 5.3 - In Exercises 1-10 decide whether each statement is...Ch. 5.3 - In Exercises 1-10 decide whether each statement is...Ch. 5.3 - Prob. 5ECh. 5.3 - In Exercises 1-10 decide whether each statement is...Ch. 5.3 - In Exercises 1-10 decide whether each statement is...Ch. 5.3 - Prob. 8ECh. 5.3 - Prob. 9ECh. 5.3 - Prob. 10ECh. 5.3 - Prob. 11ECh. 5.3 - Prob. 12ECh. 5.3 - Prob. 13ECh. 5.3 - Prob. 14ECh. 5.3 - It has been proved that the reciprocals of all the...Ch. 5.3 - Prob. 16ECh. 5.3 - Prob. 17ECh. 5.3 - Prob. 18ECh. 5.3 - Prob. 19ECh. 5.3 - Prob. 20ECh. 5.3 - 21. There are four abundant numbers between 1 and...Ch. 5.3 - Prob. 22ECh. 5.3 - Prob. 23ECh. 5.3 - Prob. 24ECh. 5.3 - 25. The proper divisors of 1184 are 1.2. 4. 8, 16,...Ch. 5.3 - Prob. 26ECh. 5.3 - Prob. 27ECh. 5.3 - Prob. 28ECh. 5.3 - Prob. 29ECh. 5.3 - Prob. 30ECh. 5.3 - Prob. 31ECh. 5.3 - Prob. 32ECh. 5.3 - Prob. 33ECh. 5.3 - Prob. 34ECh. 5.3 - Prob. 35ECh. 5.3 - Prob. 36ECh. 5.3 - Prob. 37ECh. 5.3 - Prob. 38ECh. 5.3 - The first four perfect numbers were identified in...Ch. 5.3 - Prob. 40ECh. 5.3 - Prob. 41ECh. 5.3 - Prob. 42ECh. 5.3 - Prob. 43ECh. 5.3 - Prob. 44ECh. 5.3 - Prob. 45ECh. 5.3 - Prob. 46ECh. 5.3 - 47. Explain why the primorial formula does not...Ch. 5.3 - Prob. 48ECh. 5.3 - 49. Choose the correct completion: The primorial...Ch. 5.3 - Prob. 50ECh. 5.3 - Prob. 51ECh. 5.3 - Prob. 52ECh. 5.3 - Prob. 53ECh. 5.3 - Prob. 54ECh. 5.3 - Prob. 55ECh. 5.3 - Prob. 56ECh. 5.3 - Prob. 57ECh. 5.3 - Prob. 58ECh. 5.3 - Prob. 59ECh. 5.3 - Prob. 60ECh. 5.3 - Prob. 61ECh. 5.3 - Prob. 62ECh. 5.3 - Prob. 63ECh. 5.3 - Prob. 64ECh. 5.3 - Prob. 65ECh. 5.3 - Prob. 66ECh. 5.3 - Prob. 67ECh. 5.3 - Prob. 68ECh. 5.4 - Decide whether each statement is true or false. No...Ch. 5.4 - Decide whether each statement is true or false.
2....Ch. 5.4 - Decide whether each statement is true or false. If...Ch. 5.4 - Decide whether each statement is true or false.
4....Ch. 5.4 - Decide whether each statement is true or false....Ch. 5.4 - Decide whether each statement is true or false....Ch. 5.4 - Decide whether each statement is true or false....Ch. 5.4 - Decide whether each statement is true or false....Ch. 5.4 - Decide whether each statement is true or false.
9....Ch. 5.4 - Decide whether each statement is true or false....Ch. 5.4 - Use the prime factors method to find the greatest...Ch. 5.4 - Use the prime factors method to find the greatest...Ch. 5.4 - Use the prime factors method to find the greatest...Ch. 5.4 - Use the prime factors method to find the greatest...Ch. 5.4 - Use the prime factors method to find the greatest...Ch. 5.4 - Use the prime factors method to find the greatest...Ch. 5.4 - Use the method of dividing by prime factors to...Ch. 5.4 - Use the method of dividing by prime factors to...Ch. 5.4 - Use the method of dividing by prime factors to...Ch. 5.4 - Use the method of dividing by prime factors to...Ch. 5.4 - Use the method of dividing by prime factors to...Ch. 5.4 - Use the method of dividing by prime factors to...Ch. 5.4 - Use the Euclidean algorithm to find the greatest...Ch. 5.4 - Use the Euclidean algorithm to find the greatest...Ch. 5.4 - Use the Euclidean algorithm to find the greatest...Ch. 5.4 - Use the Euclidean algorithm to find the greatest...Ch. 5.4 - Use the Euclidean algorithm to find the greatest...Ch. 5.4 - Use the Euclidean algorithm to find the greatest...Ch. 5.4 - Use the prime factors method to find the least...Ch. 5.4 - Use the prime factors method to find the least...Ch. 5.4 - Use the prime factors method to find the least...Ch. 5.4 - Use the prime factors method to find the least...Ch. 5.4 - Use the prime factors method to find the least...Ch. 5.4 - Use the prime factors method to find the least...Ch. 5.4 - Use the method of dividing by prime factors to...Ch. 5.4 - Use the method of dividing by prime factors to...Ch. 5.4 - Use the method of dividing by prime factors to...Ch. 5.4 - Use the method of dividing by prime factors to...Ch. 5.4 - Use the method of dividing by prime factors to...Ch. 5.4 - Use the method of dividing by prime factors to...Ch. 5.4 - Use the formula given in the text on page 203and...Ch. 5.4 - Use the formula given in the text on page 203 and...Ch. 5.4 - Use the formula given in the text on page 203 and...Ch. 5.4 - Use the formula given in the text on page 203 and...Ch. 5.4 - Use the formula given in the text on page 203 and...Ch. 5.4 - Use the formula given in the text on page 203 and...Ch. 5.4 - Explain in your own words how to find the greatest...Ch. 5.4 - 48, Explain in your own words how to find the...Ch. 5.4 - If p. q, and r and different primes, and a. b, and...Ch. 5.4 - Find (a) the greatest common factor and (b) the...Ch. 5.4 - Prob. 51ECh. 5.4 - Prob. 52ECh. 5.4 - Prob. 53ECh. 5.4 - It is possible to extend the Euclidean algorithm...Ch. 5.4 - Prob. 55ECh. 5.4 - Suppose that the least common multiple of p and q...Ch. 5.4 - Prob. 57ECh. 5.4 - Prob. 58ECh. 5.4 - Prob. 59ECh. 5.4 - Refer to Examples 9 and 10 to solve each problem....Ch. 5.4 - Refer to Examples 9 and 10 to solve each...Ch. 5.4 - Refer to Examples 9 and 10 to solve each...Ch. 5.4 - Prob. 63ECh. 5.4 - Refer to Examples 9 and 10 to solve each problem....Ch. 5.5 - Answer each question concerning the Fibonacci...Ch. 5.5 - Prob. 2ECh. 5.5 - Prob. 3ECh. 5.5 - Prob. 4ECh. 5.5 - Prob. 5ECh. 5.5 - Prob. 6ECh. 5.5 - Prob. 7ECh. 5.5 - Prob. 8ECh. 5.5 - Prob. 9ECh. 5.5 - Prob. 10ECh. 5.5 - Prob. 11ECh. 5.5 - Prob. 12ECh. 5.5 - Prob. 13ECh. 5.5 - Prob. 14ECh. 5.5 - Prob. 15ECh. 5.5 - It has been shown that if m divides n, then Fm is...Ch. 5.5 - Prob. 17ECh. 5.5 - Prob. 18ECh. 5.5 - Prob. 19ECh. 5.5 - Prob. 20ECh. 5.5 - Prob. 21ECh. 5.5 - Prob. 22ECh. 5.5 - Prob. 23ECh. 5.5 - Prob. 24ECh. 5.5 - Prob. 25ECh. 5.5 - Prob. 26ECh. 5.5 - Prob. 27ECh. 5.5 - Recall (lie Pythagorean theorem from geometry: If...Ch. 5.5 - Recall (lie Pythagorean theorem from geometry: If...Ch. 5.5 - Prob. 30ECh. 5.5 - Prob. 31ECh. 5.5 - Prob. 32ECh. 5.5 - Prob. 33ECh. 5.5 - Prob. 34ECh. 5.5 - Prob. 35ECh. 5.5 - Prob. 36ECh. 5 - In Exercises 1-6, decide whether each statement is...Ch. 5 - In Exercises 1-6, decide whether each statement is...Ch. 5 - Prob. 3TCh. 5 - In Exercises 1-6, decide whether each statement is...Ch. 5 - Prob. 5TCh. 5 - Prob. 6TCh. 5 - Use divisibility tests to determine whether the...Ch. 5 - Prob. 8TCh. 5 - Prob. 9TCh. 5 - Prob. 10TCh. 5 - Prob. 11TCh. 5 - Prob. 12TCh. 5 - Give a pair of twin primes between 60 and 80.Ch. 5 - Prob. 14TCh. 5 - Prob. 15TCh. 5 - Prob. 16TCh. 5 - Prob. 17TCh. 5 - Prob. 18TCh. 5 - Prob. 19TCh. 5 - Prob. 20TCh. 5 - 21. Choose any term after the first in the...Ch. 5 - 22. Which one of the following is the exact value...
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