/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 31 Prove that there is no positive ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 31

Prove that there is no positive integer \(n\) such that \(n^{2}+\) \(n^{3}=100 .\)

Short Answer

Expert verified
No positive integer n satisfies n^2 + n^3 = 100.

Step by step solution

01

Understand the Given Equation

We need to prove that there is no positive integer n such that the sum of its square and cube equals 100, i.e., a^2 + a^3 = 100.
02

Form the Equation

Rewrite the given equation as:a^2 + a^3 = 100
03

Analyze the Function

Consider the function f(a) = a^2 + a^3. This is a polynomial function.
04

Test Small Positive Integers

Calculate the values of f(a) for a few small positive integers:For a = 1: f(1) = 1^2 + 1^3 = 1 + 1 = 2For a = 2: f(2) = 2^2 + 2^3 = 4 + 8 = 12For a = 3: f(3) = 3^2 + 3^3 = 9 + 27 = 36For a = 4: f(4) = 4^2 + 4^3 = 16 + 64 = 80For a = 5: f(5) = 5^2 + 5^3 = 25 + 125 = 150
05

Conclusion

Since the function f(a) = a^2 + a^3 equals neither 100 for a = 1, 2, 3, 4, nor does it produce values below 100 for a > 4, we conclude that there is no positive integer a such that a^2 + a^3 = 100.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integer Solutions
To solve problems in discrete mathematics, determining whether equations have integer solutions is crucial. In our exercise, we are dealing with the equation: a^2 + a^3 = 100 Here, we look for integer values of 'a' that satisfy this equation. Integer solutions are values like -2, -1, 0, 1, 2, and so on. They are whole numbers, and they play a fundamental role in many areas of mathematics. To determine if there are positive integers (whole numbers greater than zero) that satisfy our equation, we need to test small positive integers. If none fit, we might conclude there are no solutions in this set.
Polynomial Functions
Polynomial functions are a type of mathematical expression involving sums of powers of variables. Our equation: a^2 + a^3 is a polynomial function of degree 3 because the highest power of 'a' is 3. These functions are smooth and continuous, which helps in predicting their behavior. When analyzing polynomial functions, we look at their values, graphs, and roots. Here, by calculating the polynomial function for small integers (e.g., 1, 2, 3, 4, and 5), we observe how the values change. This can help us understand whether our function reaches the value 100 for any integer 'a'.
Proof Techniques
Proof techniques are methods used to establish the truth or falsehood of mathematical statements. To show that a particular equation has no solutions, we often use proof by exhaustion. This method involves verifying all possible cases. In our exercise, we checked several small positive integers to see if they satisfy a^2 + a^3 = 100. For a complete integer proof, we may extend our checks or use other proof techniques like contradiction: assume a solution exists and show this leads to an impossible conclusion.
Non-Existence Proofs
Non-existence proofs demonstrate that no solution exists for a given problem under specific conditions. In our case, we aim to show there are no positive integers 'a' such that a^2 + a^3 = 100. After checking small values of 'a' and observing that none satisfy the equation, and realizing that the function's values do not hit 100 beyond a point (as larger values exceed 100), we conclude that no such 'a' exists. Thus, we provide a non-existence proof, proving that our equation has no solution in the set of positive integers.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Construct a compound proposition that asserts that every cell of a \(9 \times 9\) Sudoku puzzle contains at least one number.

Use quantifiers and logical connectives to express the fact that a quadratic polynomial with real number coefficients has at most two real roots.

For each of these collections of premises, what relevant conclusion or conclusions can be drawn? Explain the rules of inference used to obtain each conclusion from the premises. a) 鈥淚f I take the day off, it either rains or snows.鈥� 鈥淚 took Tuesday off or I took Thursday off.鈥� 鈥淚t was sunny on Tuesday.鈥� 鈥淚t did not snow on Thursday.鈥� b) 鈥淚f I eat spicy foods, then I have strange dreams.鈥� 鈥淚 have strange dreams if there is thunder while I sleep.鈥� 鈥淚 did not have strange dreams.鈥� c) 鈥淚 am either clever or lucky.鈥� 鈥淚 am not lucky.鈥� 鈥淚f I am lucky, then I will win the lottery.鈥� d) 鈥淓very computer science major has a personal computer.鈥� 鈥淩alph does not have a personal computer.鈥� 鈥淎nn has a personal computer.鈥� e) 鈥淲hat is good for corporations is good for the United States.鈥� 鈥淲hat is good for the United States is good for you.鈥� 鈥淲hat is good for corporations is for you to buy lots of stuff.鈥� f ) 鈥淎ll rodents gnaw their food.鈥� 鈥淢ice are rodents.鈥� 鈥淩abbits do not gnaw their food.鈥� 鈥淏ats are not ro- dents.鈥�

Suppose that five ones and four zeros are arranged around a circle. Between any two equal bits you insert a 0 and between any two unequal bits you insert a 1 to produce nine new bits. Then you erase the nine original bits. Show that when you iterate this procedure, you can never get nine zeros. [Hint: Work backward, assuming that you did end up with nine zeros.]

Suppose that \(a\) and \(b\) are odd integers with \(a \neq b .\) Show there is a unique integer \(c\) such that \(|a-c|=|b-c|\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.