Problem 25
Question
Prove by cases, where \(n\) is an arbitrary integer and \(|x|\) denotes the absolute value of \(x\). \(n^{3}-n\) is divisible by \(3 .\) (Hint: Assume that every integer is of the form \(3 k, 3 k+1, \text { or } 3 k+2 .)\)
Step-by-Step Solution
Verified Answer
In all three cases, we showed that the expression \(n^3 - n\) is divisible by \(3\). For \(n=3k, (3k)^3 - 3k = 3k(9k^2 - 1)\) is divisible by 3. For \(n=3k+1, (3k + 1)^3 - (3k + 1) = 3(9k^3 + 9k^2 + 2k)\) is divisible by 3. Finally, for \(n=3k+2, (3k + 2)^3 - (3k + 2) = 3(9k^3 + 18k^2 + 11k + 2)\) is divisible by 3. Thus, we've proven by cases that \(n^3 - n\) is divisible by \(3\) for any integer \(n\).
1Step 1: Case 1 - n = 3k
Let's assume \(n = 3k\), where \(k\) is an integer. Then we have to prove that \((3k)^3 - 3k\) is divisible by \(3\).
Substitute \(n=3k\):
\((3k)^3 - 3k = 27k^3 - 3k\)
Factor out \(3k\):
\(3k(9k^2 - 1)\)
Since \(3k\) is divisible by \(3\), the expression \((3k)^3 - 3k\) is also divisible by \(3\).
2Step 2: Case 2 - n = 3k+1
Now, let's assume \(n = 3k+1\), where \(k\) is an integer. Then we have to prove that \(((3k+1))^3 - (3k+1)\) is divisible by \(3\).
Substitute \(n=3k+1\):
\((3k + 1)^3 - (3k + 1)\)
Expand the cube \((3k + 1)^3\):
\(27k^3 + 27k^2 + 9k + 1 - 3k - 1\)
Simplify the expression:
\(27k^3 + 27k^2 + 6k\)
Factor out \(3\):
\(3(9k^3 + 9k^2 + 2k)\)
Since the expression is a multiple of \(3\), \(((3k+1))^3 - (3k+1)\) is divisible by \(3\).
3Step 3: Case 3 - n = 3k+2
Finally, let's assume \(n = 3k+2\), where \(k\) is an integer. Then we have to prove that \(((3k+2))^3 - (3k+2)\) is divisible by \(3\).
Substitute \(n=3k+2\):
\((3k + 2)^3 - (3k + 2)\)
Expand the cube \((3k + 2)^3\):
\(27k^3 + 54k^2 + 36k + 8 - 3k - 2\)
Simplify the expression:
\(27k^3 + 54k^2 + 33k + 6\)
Factor out \(3\):
\(3(9k^3 + 18k^2 + 11k + 2)\)
Since the expression is a multiple of \(3\), \(((3k+2))^3 - (3k+2)\) is divisible by \(3\).
4Step 4: Conclusion
In each case, for \(n = 3k, 3k+1,\) and \(3k+2\), we showed that the expression \(n^3 - n\) is divisible by \(3\). Therefore, it has been proven by cases that \(n^3 - n\) is divisible by \(3\) for any integer \(n\).
Key Concepts
Proof by CasesNumber TheoryDivisibilityInteger Properties
Proof by Cases
Proof by cases is a classical method in mathematics used to demonstrate that a statement holds true for all possible situations, usually by examining a number of distinct cases separately. This technique is especially useful when a problem can be broken down into different, independent instances, each of which can be handled individually.
For the problem at hand, our task is to prove that the expression \(n^3 - n\) is divisible by 3 for any integer \(n\). To accomplish this, we use proof by cases based on the properties of integers and divisibility. By considering three distinct cases, namely when \(n\) is of the form \(3k\), \(3k+1\), and \(3k+2\), we analyze whether the condition holds true in each situation. This division into cases originates from the fact that integers divided by 3 can leave one of these specific remainders. Throughout each case, we simplify the expression \(n^3 - n\) to demonstrate its divisibility by three by factoring or other algebraic manipulations.
This approach ensures a comprehensive check of all possibilities, making it a powerful proof technique.
For the problem at hand, our task is to prove that the expression \(n^3 - n\) is divisible by 3 for any integer \(n\). To accomplish this, we use proof by cases based on the properties of integers and divisibility. By considering three distinct cases, namely when \(n\) is of the form \(3k\), \(3k+1\), and \(3k+2\), we analyze whether the condition holds true in each situation. This division into cases originates from the fact that integers divided by 3 can leave one of these specific remainders. Throughout each case, we simplify the expression \(n^3 - n\) to demonstrate its divisibility by three by factoring or other algebraic manipulations.
This approach ensures a comprehensive check of all possibilities, making it a powerful proof technique.
Number Theory
Number theory is a field of mathematics dealing with the properties and relationships of numbers, particularly integers. It involves various topics like divisibility, prime numbers, and integer solutions. This field is fundamental because it provides the building blocks for more complex mathematical concepts and theories.
In this specific problem, we explore divisions made with the number 3. Recognizing patterns in numbers, their behaviors, especially under different arithmetic operations, and their divisibility is a core part of number theory. The integers chosen in different forms such as \(3k\), \(3k+1\), and \(3k+2\) epitomize a fundamental understanding that any integer can be expressed in these forms based on the division remainder when divided by 3 — a concept extensively used in number theory.
Engaging with number theory involves techniques and methods to identify these relationships and apply them to problem-solving in an accurate and logical manner.
In this specific problem, we explore divisions made with the number 3. Recognizing patterns in numbers, their behaviors, especially under different arithmetic operations, and their divisibility is a core part of number theory. The integers chosen in different forms such as \(3k\), \(3k+1\), and \(3k+2\) epitomize a fundamental understanding that any integer can be expressed in these forms based on the division remainder when divided by 3 — a concept extensively used in number theory.
Engaging with number theory involves techniques and methods to identify these relationships and apply them to problem-solving in an accurate and logical manner.
Divisibility
Divisibility refers to the ability of one integer to divide another without leaving a remainder. This topic is essential in understanding the structure of numbers and their interactions. It is often used in proofs to establish whether one number can "fit inside" another entirely.
In our mathematical proof, divisibility by 3 is the focal point. By translating different forms of \(n\) into algebraic expressions and factoring them, we demonstrated that each form, when substituted into \(n^3-n\), results in expressions clearly divisiable by 3. For instance, when \(n = 3k+1\), after expanding and reorganizing, the expression becomes \(3(9k^3 + 9k^2 + 2k)\), which is visibly divisible by 3 due to the factor \(3\). This step confirms the mathematical requirement we set out to prove.
Focusing on the concept of divisibility assists in unveiling the inherent relationships of numbers and is a crucial skill used throughout various mathematical disciplines.
In our mathematical proof, divisibility by 3 is the focal point. By translating different forms of \(n\) into algebraic expressions and factoring them, we demonstrated that each form, when substituted into \(n^3-n\), results in expressions clearly divisiable by 3. For instance, when \(n = 3k+1\), after expanding and reorganizing, the expression becomes \(3(9k^3 + 9k^2 + 2k)\), which is visibly divisible by 3 due to the factor \(3\). This step confirms the mathematical requirement we set out to prove.
Focusing on the concept of divisibility assists in unveiling the inherent relationships of numbers and is a crucial skill used throughout various mathematical disciplines.
Integer Properties
Integer properties cover a range of characteristics that belong to whole numbers, including concepts such as evenness, oddness, and specific devised patterns from numerical operations. Integral to this collection are how numbers behave under basic arithmetic functions — addition, multiplication, etc.
The integer property utilized in our proof process is the structure of integers in mod 3 representation. This is represented by setting integers in forms such as \(3k\), \(3k+1\), or \(3k+2\) where \(k\) belongs to the set of all integers. These forms help segment the integers in a manner that simplifies checking their properties with respect to modular arithmetic, particularly useful when examining divisibility conditions.
Understanding these properties allows for a smooth and effective application of foundational number theory principles in solving abstract mathematical problems. These properties are crucial when leveraging integer behaviors for proof and verification within mathematical contexts.
The integer property utilized in our proof process is the structure of integers in mod 3 representation. This is represented by setting integers in forms such as \(3k\), \(3k+1\), or \(3k+2\) where \(k\) belongs to the set of all integers. These forms help segment the integers in a manner that simplifies checking their properties with respect to modular arithmetic, particularly useful when examining divisibility conditions.
Understanding these properties allows for a smooth and effective application of foundational number theory principles in solving abstract mathematical problems. These properties are crucial when leveraging integer behaviors for proof and verification within mathematical contexts.
Other exercises in this chapter
Problem 24
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