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Chapter 3: Problem 11

Let \(n\) be a natural number. Prove each of the following: (a) For every integer \(a, a \equiv a(\bmod n)\). This is called the reflexive property of congruence modulo \(n\). (b) For all integers \(a\) and \(b,\) if \(a \equiv b(\bmod n),\) then \(b \equiv a(\bmod n)\). This is called the symmetric property of congruence modulo \(n\). (c) For all integers \(a, b,\) and \(c,\) if \(a \equiv b(\bmod n)\) and \(b \equiv c(\bmod n),\) then \(a \equiv c(\bmod n)\) This is called the transitive property of congruence modulo \(n\).

Short Answer

Expert verified
To prove the reflexive, symmetric, and transitive properties of congruence modulo \(n\), we use the definition of congruence modulo \(n\): two integers \(a\) and \(b\) are congruent modulo \(n\) if \(n\) divides their difference \((a-b)\). (a) The reflexive property holds because every natural number \(n\) divides \((a-a)=0\). (b) For the symmetric property, if \(a \equiv b(\bmod n)\), then \(n\) divides \((a-b)\) and there exists an integer \(k\) such that \(a - b = kn\). Multiplying both sides by \(-1\), we get \((b-a)=-kn\). This shows that \(n\) divides \((b-a)\), proving that \(b \equiv a(\bmod n)\). (c) For the transitive property, given \(a \equiv b(\bmod n)\) and \(b \equiv c(\bmod n)\), we know that \(n\) divides \((a-b)\) and \(n\) divides \((b-c)\). If there exist integers \(k_1\) and \(k_2\) such that \((a-b)=k_1\cdot n\) and \((b-c)=k_2\cdot n\), then the sum \((a-b)+(b-c)=(a-c)\) can be written as \((k_1+k_2)n\). Since \(k_1+k_2\) is an integer, \(n\) divides \((a-c)\), proving the transitive property \(a \equiv c(\bmod n)\).

Step by step solution

01

Proving Reflexive Property

To prove that \(a \equiv a(\bmod n)\) for every integer \(a\), we need to show that \(n\) divides \((a-a)\). Let's examine the difference of \(a\) and \(a\): \((a-a)=0\) Since every natural number \(n\) divides \(0\), the reflexive property holds true for all integers \(a\).
02

Proving Symmetric Property

To prove the symmetric property, we assume that \(a \equiv b(\bmod n)\), and we need to show that \(b \equiv a(\bmod n)\). By the definition of congruence modulo \(n\), we know that \(a \equiv b(\bmod n)\) means that \(n\) divides \((a-b)\). Let's suppose there exists a integer \(k\) such that \((a-b) = k\cdot n\), then we can rewrite the equation as \((a-b) = n\cdot k\). Now, if we multiply both sides of the equation by \((-1)\), we get: \(-(a-b) = -n\cdot k\) \((b-a) = n\cdot(-k)\) Now we have shown that \(n\) divides \((b-a)\), which means \(b \equiv a(\bmod n)\). This proves the symmetric property of congruence modulo \(n\).
03

Proving Transitive Property

To prove the transitive property, we are given \(a \equiv b(\bmod n)\) and \(b \equiv c(\bmod n)\), and we need to show that \(a \equiv c(\bmod n)\). Let's examine the differences: Since \(a \equiv b(\bmod n)\), we know \(n\) divides \((a-b)\). Let's assume there exists an integer \(k_1\) such that \((a-b) = k_1\cdot n\). Since \(b \equiv c(\bmod n)\), we also know that \(n\) divides \((b-c)\). Let's assume there exists an integer \(k_2\) such that \((b-c) = k_2\cdot n\). Now, we add \((a-b)+(b-c)\): \((a-b)+(b-c)= (a-c)\) From the previous equations, \((a-b) = k_1\cdot n\) and \((b-c) = k_2\cdot n\). Let's plug these into our sum: \(k_1\cdot n + k_2\cdot n = (a-c)\) \((k_1+k_2)\cdot n = (a-c)\) Since \((k_1+k_2)\) is an integer, \((a-c)\) can be written as a multiple of \(n\). Therefore, \(n\) divides \((a-c)\). We have shown that if \(a \equiv b(\bmod n)\) and \(b \equiv c(\bmod n)\), then \(a \equiv c(\bmod n)\), which proves the transitive property of congruence modulo \(n\).

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Key Concepts

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

Reflexive Property of Congruence Modulo n
Understanding the reflexive property of congruence modulo n is like realizing that every number is always in harmony with itself in the world of modular arithmetic. This fundamental property is what states that any integer is always congruent to itself modulo any natural number. To put it simply, if you take an integer and compare it to itself through the lens of modular arithmetic, they will always match up perfectly, just like your reflection looking back at you from a mirror. This property serves as the building block for more complex modular arithmetic proofs and computations. By recognizing that any integer's self-congruence is a given, students can move forward with confidence in solving modular equations and understanding their symmetries.
Symmetric Property of Congruence Modulo n
Much like the principle that friendship is a two-way street, the symmetric property of congruence modulo n tells us that the relationship between two congruent numbers is mutual. If integer A is related to integer B in the context of a specific modulus, then B is equally related to A. This is a powerful idea because it means that congruence isn't a one-sided affair; it's a balanced relationship that holds equally true in both directions. Reiterating the proof of this property, which involves simple algebraic manipulation and a fundamental understanding of divisibility, reinforces the students' grasp of modular relationships. It undeniably establishes that congruence is an egalitarian concept, further paving the way for learners to comprehend its application in more complex number theory problems.
Transitive Property of Congruence Modulo n
The transitive property in arithmetic is similar to a chain reaction鈥攐nce set into motion, it carries through a sequence of events. In the realm of congruence modulo n, this property says that if one number is congruent to a second, and the second is congruent to a third, then the first and third numbers are congruent as well. By connecting the dots between the proofs of the reflexive and symmetric properties, the transitive property cements our understanding of the stable and predictable nature of modular arithmetic. This concept is akin to a mathematical 'trust exercise,' reassuring us that if the first two relationships hold, we can count on the third to be true as well, providing a robust framework for proving more involved theorems and solving compound modular equations.
Number Theory
At the heart of congruence modulo n lies number theory, a field of pure mathematics devoted entirely to the properties and relationships of numbers, especially integers. Number theory is sometimes referred to as the 'queen of mathematics' for its foundational and elegant qualities. When we delve into properties of congruences, we are in fact exploring the tip of a huge mathematical iceberg. This field covers prime numbers, the greatest common divisors, and more abstract concepts such as groups and rings, which provide the framework for understanding patterns and solving some of the most intricate puzzles in mathematics. For learners, appreciating the basics of number theory is crucial for a deep understanding of congruence and its applications in cryptography, computer science, and even in reasoning about the natural world.
Mathematical Proof
A mathematical proof is a rigorous and logical process used to demonstrate the truth of a mathematical statement. It's essentially the bedrock of all mathematical knowledge, allowing mathematicians to establish with certainty whether certain propositions are true. In the context of congruence modulo n, each property proved serves as a stepping stone that lays the groundwork for higher levels of mathematical reasoning. Proofs often employ a variety of techniques such as direct proof, proof by contradiction, and induction. Familiarizing students with proofs not only strengthens their logic and reasoning skills but also instills an appreciation for the precision and elegance with which mathematics can describe and explain the world. By learning to construct and follow proofs step by step, students gain the tools to navigate more advanced mathematical concepts with clarity and confidence.

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Most popular questions from this chapter

Is the following proposition true or false? Justify your conclusion with a counterexample or a proof. For each integer \(a, 3\) divides \(a^{3}+23 a\).

Using only the digits 1 through 9 one time each, is it possible to construct a 3 by 3 magic square with the digit 3 in the center square? That is, is it possible to construct a magic square of the form $$ \begin{array}{|c|c|c|} \hline a & b & c \\ \hline d & 3 & e \\ \hline f & g & h \\ \hline \end{array} $$ where \(a, b, c, d, e, f, g, h\) are all distinct digits, none of which is equal to 3 ? Either construct such a magic square or prove that it is not possible.

Prove the following proposition: For all integers \(a\) and \(b\), if 3 divides \(\left(a^{2}+b^{2}\right)\), then 3 divides \(a\) and 3 divides \(b\).

Evaluation of Proofs See the instructions for Exercise (19) on page 100 from Section 3.1 . (a) Proposition. If \(m\) is an odd integer, then \((m+6)\) is an odd integer. Proof. For \(m+6\) to be an odd integer, there must exist an integer \(n\) such that $$ m+6=2 n+1 $$ By subtracting 6 from both sides of this equation, we obtain $$ \begin{aligned} m &=2 n-6+1 \\ &=2(n-3)+1 . \end{aligned} $$ By the closure properties of the integers, \((n-3)\) is an integer, and hence, the last equation implies that \(m\) is an odd integer. This proves that if \(m\) is an odd integer, then \(m+6\) is an odd integer. (b) Proposition. For all integers \(m\) and \(n,\) if \(m n\) is an even integer, then \(m\) is even or \(n\) is even. \mathrm{\\{} \text { Proof } . For either \(m\) or \(n\) to be even, there exists an integer \(k\) such that \(m=2 k\) or \(n=2 k .\) So if we multiply \(m\) and \(n,\) the product will contain a factor of 2 and, hence, \(m n\) will be even.

(a) Prove that for each integer \(a\), if \(a \neq 0(\bmod 7),\) then \(a^{2} \neq 0(\bmod 7)\). (b) Prove that for each integer \(a\), if 7 divides \(a^{2}\), then 7 divides \(a\). (c) Prove that the real number \(\sqrt{7}\) is an irrational number.
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