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Chapter 3: Problem 7

Let \(a\) be an integer and let \(n \in \mathbb{N}\). (a) Prove that if \(a \equiv 0(\bmod n),\) then \(n \mid a\). (b) Prove that if \(n \mid a\), then \(a \equiv 0(\bmod n)\).

Short Answer

Expert verified
We proved the two statements as follows: 1. If \(a \equiv 0(\bmod n)\), then \(n \mid a\). This is because the difference between \(a\) and \(0\) is \(a\), which is divisible by \(n\), leading to \(n \mid a\). 2. If \(n \mid a\), then \(a \equiv 0(\bmod n)\). This is because if \(n\) divides \(a\) exactly, there exists an integer \(k\) such that \(a = n \cdot k\). As a result, the difference between \(a\) and \(0\) is divisible by \(n\), leading to \(a \equiv 0(\bmod n)\).

Step by step solution

01

Definition of congruence modulo n

To prove this statement, we first need to know the definition of congruence modulo n: Two integers \(a\) and \(b\) are congruent modulo n, if their difference \(a - b\) is divisible by \(n\). It is written as \(a \equiv b(\bmod n)\). Now, we are given that \(a \equiv 0(\bmod n)\). Based on the definition above, this means that the difference \(a - 0 = a\) is divisible by \(n\).
02

Proving divisibility

Since we know that the difference \(a\) is divisible by \(n\), we can write this as: $$n \mid (a - 0) \implies n \mid a$$ Thus, we have proved that if \(a \equiv 0(\bmod n)\), then \(n \mid a\). ##Part (b): Proving if \(n \mid a\), then \(a \equiv 0(\bmod n)\)##
03

Definition of divisibility

To prove this statement, we first need to know the definition of divisibility: If an integer \(n\) divides another integer \(a\) exactly with no remainder, it is written as \(n \mid a\). In other words, there exists an integer \(k\) such that \(a = n \cdot k\). Now, we are given that \(n \mid a\). This implies that there exists an integer \(k\) such that \(a = n \cdot k\).
04

Proving congruence modulo n

We have \(a = n \cdot k\). Now, subtract \(0\) from both sides: $$a - 0 = n \cdot k$$ We can see that the difference, \(a - 0\), is divisible by \(n\). Based on the definition of congruence modulo n, this can be written as: $$a \equiv 0(\bmod n)$$ Thus, we have proved that if \(n \mid a\), then \(a \equiv 0(\bmod n)\). In conclusion, the two statements are proved as follows: 1. If \(a \equiv 0(\bmod n)\), then \(n \mid a\). 2. If \(n \mid a\), then \(a \equiv 0(\bmod n)\).

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Key Concepts

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

Divisibility in Mathematics
Divisibility is a concept in mathematics that allows us to determine if one number can be evenly divided by another without leaving a remainder. When an integer n divides another integer a, this is represented by the notation \( n \mid a \). This notation translates to saying that n is a divisor of a, or a is divisible by n.

For instance, if we have the integers 6 and 24, we can say that 6 divides 24 because when 24 is divided by 6, the result is exactly 4, without any remainder. In mathematical terms, this is written as \( 6 \mid 24 \).

Understanding divisibility is crucial when working with modular congruence because it helps us establish the rules of the arithmetic under a modular system. In the context of the exercise where we examine the relationship between modulus operation and divisibility, the notation expands to a fundamental property: if \( a \equiv 0(\bmod n) \), it implies that n divides a completely.
Proof Writing
Proof writing is a formal process used to establish the truth of a mathematical statement. When writing a proof, it is vital to start from known definitions, axioms, or previously established theorems, and then use logical reasoning to arrive at the proposition's conclusion.

In the context of proving modular congruences and divisibility, proof writing involves stating clearly the definition of congruence modulo n and the condition of divisibility.

Clear Definitions and Logical Steps

For instance, to prove that \( n \mid a \) implies \( a \equiv 0(\bmod n) \), we need to define what it means for an integer to divide another integer. After establishing the definitions, we apply them by showing that the existence of an integer k for which \( a = n \cdot k \) leads logically to the conclusion that the difference \( a - 0 \) is divisible by n, establishing our desired congruence.

Proof writing is not just a demonstration of the truth but also a communication tool to convey complex ideas clearly and logically, making it easier for others to follow and understand the reasoning. This process enhances critical thinking and the ability to construct and dissect mathematical arguments.
Integers and Natural Numbers
Integers and natural numbers are foundational concepts in mathematics. Integers, represented by the symbol \( \mathbb{Z} \), include all the whole numbers and their negatives; for example, -3, -2, -1, 0, 1, 2, 3, and so on. On the other hand, natural numbers, denoted by \( \mathbb{N} \), are a subset of integers that include all positive integers starting from 1: 1, 2, 3, and so forth.

It's relevant to note that natural numbers are always used to define modulus or modularity as they represent the set of possible remainders upon division.

Role in Modular Arithmetic

In modular arithmetic, we always use a positive integer n (which is a natural number) as the modulus. The choice of n defines the system of congruences because it limits the set of potential remainders after division. In the given exercise, n belongs to \( \mathbb{N} \), emphasizing that the modulus in a congruence relation must be a natural number.

Understanding the difference and the particular use of integers and natural numbers is crucial when dealing with modular arithmetic and proofs involving congruence and divisibility because it sets the correct framework for these operations and logical arguments.

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Most popular questions from this chapter

(Exercise (15), Section 3.1) Let \(r\) be a positive real number. The equation for a circle of radius \(r\) whose center is the origin is \(x^{2}+y^{2}=r^{2}\). (a) Use implicit differentiation to determine \(\frac{d y}{d x}\). (b) (Exercise (17), Section 3.2) Let \((a, b)\) be a point on the circle with \(a \neq 0\) and \(b \neq 0\). Determine the slope of the line tangent to the circle at the point \((a, b)\). (c) Prove that the radius of the circle to the point \((a, b)\) is perpendicular to the line tangent to the circle at the point \((a, b)\). Hint: Two lines (neither of which is horizontal) are perpendicular if and only if the products of their slopes is equal to -1

Consider the following proposition: For each integer \(a, a \equiv 3(\bmod 7)\) if and only if \(\left(a^{2}+5 a\right) \equiv 3(\bmod 7)\). (a) Write the proposition as the conjunction of two conditional statements. (b) Determine if the two conditional statements in Part (a) are true or false. If a conditional statement is true, write a proof, and if it is false, provide a counterexample. (c) Is the given proposition true or false? Explain.

Is the following statement true or false? Justify your conclusion. For each integer \(n\) that is greater than 1 , if \(a\) is the smallest positive factor of \(n\) that is greater than \(1,\) then \(a\) is prime. See Exercise (13) in Section 2.4 (page 78 ) for the definition of a prime number and the definition of a composite number.

Are the following statements true or false? Justify your conclusions. (a) For all integers \(a\) and \(b,(a+b)^{2} \equiv\left(a^{2}+b^{2}\right)(\bmod 2)\). (b) For all integers \(a\) and \(b,(a+b)^{3} \equiv\left(a^{3}+b^{3}\right)(\bmod 3)\). (c) For all integers \(a\) and \(b,(a+b)^{4} \equiv\left(a^{4}+b^{4}\right)(\bmod 4)\). (d) For all integers \(a\) and \(b,(a+b)^{5} \equiv\left(a^{5}+b^{5}\right)(\bmod 5)\). If any of the statements above are false, write a new statement of the following form that is true (and prove that it is true): For all integers \(a\) and \(b,(a+b)^{n} \equiv\left(a^{n}+\right.\) something \(\left.+b^{n}\right)(\bmod n)\).

In Section \(3.1,\) we defined congruence modulo \(n\) where \(n\) is a natural number. If \(a\) and \(b\) are integers, we will use the notation \(a \neq b(\bmod n)\) to mean that \(a\) is not congruent to \(b\) modulo \(n\). * (a) Write the contrapositive of the following conditional statement: For all integers \(a\) and \(b,\) if \(a \neq 0(\bmod 6)\) and \(b \neq 0(\bmod 6),\) then \(a b \not \equiv 0(\bmod 6)\). (b) Is this statement true or false? Explain.
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