Question

It is straightforward to solve for \(x\) any equation of the form $$ x+_{n} a=b $$ in \(Z_{n}\) and to see that the result will be a unique value of \(x\). However, in the discussion of Exercise 2.1-6, we saw that \(0,3,6\), and 9 are all solutions to the equation $$ 4 \cdot{ }_{12} x=0 $$ a. Are there any integral values of \(a\) and \(b\), with 1 less than or equal to \(a\) and \(b\), which are both less than 12, for which the equation \(a \cdot{ }_{12} x=b\) does not have any solutions in \(Z_{12}\) ? If there are, give one set of values for \(a\) and \(b\). If there are not, explain how you know this. b. Are there any integers \(a\), with \(1

Short answer

a. Example: \( a = 2, b = 3 \), no solution. b. \( a = 5 \) works for every \( b \).

Step-by-step solution

Understanding Modular Arithmetic

We are dealing with equations in the modular system \( Z_{12} \), which means all arithmetic is done modulo 12. An equation \( a \cdot_{12} x = b \) is solved by finding \( x \) such that when \( ax \) is calculated, it is equal to \( b \) modulo 12.

Finding Conditions for Existence of Solutions

The equation \( a \cdot_{12} x = b \) has solutions if and only if the greatest common divisor (GCD) of \( a \) and 12 divides \( b \). This is because solutions exist when there is a way to express \( b \) as a multiple of \( a \), modulo 12.

Checking for No Solutions (Part A)

To find a scenario where no solutions exist, find \( a \) and \( b \) such that \( \gcd(a,12) \) does not divide \( b \). For example, if \( a = 2 \) and \( b = 3 \), since \( \gcd(2,12) = 2 \), and 2 does not divide 3, there are no solutions.

Identifying Always-Solution Scenario (Part B)

For an integer \( a \) to ensure a solution exists for every \( b \), \( a \) must be coprime with 12 (i.e., \( \gcd(a,12) = 1 \)). This means \( a \) is not a divisor nor shares common factors with 12 besides 1. One such number is \( a = 5 \). For every \( b \) that is coprime, \( 5 \cdot_{12} x=b \) has solutions because 5's inverse exists in \( Z_{12} \).

Key concepts

Greatest Common Divisor

The greatest common divisor (GCD) is a fundamental concept in number theory, which helps us understand important aspects of modular systems. When we talk about the GCD of two numbers, we're referring to the largest number that can exactly divide both of them. In simpler terms, it's the biggest number that they both have as a factor. The GCD is especially crucial when solving equations in modular arithmetic, like in the modular systems used in exercises such as the one with modulo 12.In a modular system, if you're given an equation like \(a \cdot_{12} x = b\), it can only be solved if the GCD of \(a\) and the modulus, which is 12 in this case, divides \(b\) exactly. Let's break this down: If \(\gcd(a, 12)\) divides \(b\), you can find a value for \(x\) that satisfies the equation. Otherwise, if \(\gcd(a, 12)\) does not divide \(b\), there is no solution.For example, consider \(a = 2\) and \(b = 3\). The GCD of 2 and 12 is 2. However, since 2 does not divide 3, this equation would not have a solution in \(Z_{12}\). Thus, understanding the GCD helps us determine the feasibility of finding solutions in modular arithmetic.

Unique Solutions in Modular Systems

When dealing with modular systems, one important property is determining when solutions to equations are unique. A solution is unique in a modular arithmetic system if there is exactly one specific value that satisfies the equation within the given modular set.In an equation like \(a \cdot_{n} x = b\), a unique solution exists if the number \(a\) has an inverse in the modular system \(Z_{n}\). The existence of this inverse is dependent on the relationship between \(a\) and \(n\), often determined by their GCD.If \(\gcd(a, n) = 1\), this means \(a\) and \(n\) are coprime, and an inverse exists. As a result, there will be a unique solution for every \(b\) in the modular system \(Z_{n}\). The uniqueness comes from the fact that each value of \(x\) that solves \(a \cdot_{n} x = b\) can only appear once in any given cycle of the modulo.Let's consider an example with \(a = 5\) and \(n = 12\) in the system \(Z_{12}\). Since \(\gcd(5, 12) = 1\), 5 has an inverse in \(Z_{12}\), allowing \(5 \cdot_{12} x = b\) to have a unique solution for each \(b\). Identifying the conditions for unique solutions is a key step in modular arithmetic problems.

Coprime Numbers

Coprime numbers play a vital role in modular arithmetic, determining the existence of solutions in equations within modular systems. Two numbers are considered coprime if their greatest common divisor (GCD) is 1. This means they have no shared factors other than 1, showcasing an independent and distinct relationship between them.In modular systems, if the number \(a\) is coprime with the modulus \(n\), then \(a\) has an inverse in the system \(Z_{n}\). This property is crucial because it ensures the equation \(a \cdot_{n} x = b\) has at least one solution for every \(b\) in the system. This is because the inverse allows us to "undo" the multiplication by \(a\), effectively solving for \(x\).For example, returning to the equation with \(a = 5\) and modulus \(n = 12\), \(5\) and \(12\) are coprime. Therefore, \(5\) has an inverse, ensuring that there is a solution for every \(b\) in \(Z_{12}\). Coprime numbers thus assure us of the presence of solutions in diverse equations, being a cornerstone for many exercises involving modular arithmetic. Understanding coprime relationships helps draw the link between number theory and practical solutions in modular systems.