Question

Find all \(a\) in \(\mathbb{Z}_{10}\) such that: a. \(a^{2}=a\) b. \(a\) has an inverse (and find the inverse). c. \(a^{k}=0\) for some \(k \geq 1\). d. \(a=2^{k}\) for some \(k \geq 1\). e. \(a=b^{2}\) for some \(b\) in \(\mathbb{Z}_{10}\).

Short answer

a = 0, 1 for (a); a = 1, 3, 7, 9 for (b); a = 0 for (c); a = 2, 4, 6, 8 for (d); a = 0, 1, 4, 5, 6, 9 for (e).

Step-by-step solution

Find solutions for a² = a

In modulo 10, solve the equation \(a^2 = a\). This simplifies to \(a(a - 1) = 0\). Since we are working modulo 10, \(a\) must be such that \(a(a-1)\equiv0 \pmod{10}\). Hence, the possible solutions are \(a = 0\) or \(a = 1\).

Find elements with inverses

In order for an integer \(a\) in \(\mathbb{Z}_{10}\) to have a multiplicative inverse, \(a\) must be coprime to 10. Check each element 0 through 9 to determine which are coprime: 1, 3, 7, and 9. Their respective inverses can be found using the extended Euclidean algorithm: \(1^{-1} = 1\), \(3^{-1} = 7\), \(7^{-1} = 3\), \(9^{-1} = 9\).

Find solutions for a^k = 0

Find all \(a\) such that \(a^k \equiv 0 \pmod{10}\) for some \(k \geq 1\). Only \(a = 0\) satisfies \(a^k \equiv 0\) as \(0^k = 0\) for any \(k \geq 1\).

Find solutions for a = 2^k

For \(a = 2^k\) in \(\mathbb{Z}_{10}\), compute powers of 2: \(2^1 \equiv 2\), \(2^2 \equiv 4\), \(2^3 \equiv 8\), \(2^4 \equiv 6\), \(2^5 \equiv 2\) (and it cycles through now). Thus, possible values of \(a\) are 2, 4, 6, and 8.

Find solutions for a = b^2

Find \(b\) such that \(b^2 \equiv a \pmod{10}\). The squares are: \(0^2 = 0\), \(1^2 = 1\), \(2^2 = 4\), \(3^2 = 9\), \(4^2 = 6\), \(5^2 = 5\), \(6^2 = 6\), \(7^2 = 9\), \(8^2 = 4\), \(9^2 = 1\). The possible values for \(a\) are 0, 1, 4, 5, 6, and 9.

Key concepts

Inverse Elements in Modular Arithmetic

Understanding inverse elements in modular arithmetic can be quite intriguing. An inverse element for a number \( a \) in modular arithmetic exists if there is another number \( b \) such that their product is congruent to 1 modulo \( n \). Essentially, \( a \cdot b \equiv 1 \pmod{n} \).

In

• this article, we consider the modular arithmetic over \( \mathbb{Z}_{10} \), which means we are working with numbers 0 through 9.
• The key requirement for an inverse is that the number must be coprime to the modulus (in this case, 10).
• This means the number shares no common factors with 10 other than 1.

Taking the elements of \( \mathbb{Z}_{10} \), it's found that:

• \(1\), \(3\), \(7\), and \(9\) are coprime with \(10\).
• Their respective inverses, calculated using techniques like the extended Euclidean algorithm, are \(1\), \(7\), \(3\), and \(9\) respectively.

This insight reveals a fundamental aspect of modular arithmetic and how it can be applied to find inverses within different modulus systems.

Congruences

The concept of congruences is central in modular arithmetic. A congruence relation means that two numbers leave the same remainder when divided by a particular modulus. We express this by saying \( a \equiv b \pmod{n} \).

Consider the scenario

• where you need to solve quadratic congruences such as \( a^2 \equiv a \pmod{10} \).
• Simplifying, we have \( a(a - 1) \equiv 0 \pmod{10} \).
• This equation is satisfied if \( a(a-1) \) is divisible by 10, leading us to the solutions \( a = 0 \) or \( a = 1 \).

The beauty of congruences is that they allow us to work with divisions and multiplications without having to solve lengthy computations directly involving large numbers or complex calculations. They help in understanding and solving modular equations efficiently.

Powers in Modular Arithmetic

Exploring powers in modular arithmetic demonstrates how numbers behave under repeated multiplication. When raising a number to higher powers, the result is reduced modulo \( n \). This process can reveal interesting patterns and cycles.

Using the powers of

• 2 as an example within \( \mathbb{Z}_{10} \):
• \(2^1 = 2\), \(2^2 = 4\), \(2^3 = 8\), \(2^4 = 6\), and\( 2^5 \equiv 2 \pmod{10}\).
• Notice the cycle repeating when the fifth power shows the same remainder as the first power, starting the cycle anew.

This cyclical nature is crucial, as it helps in simplifying power calculations and understanding modular patterns. The analysis of powers in modular arithmetic goes beyond simple calculations; it helps in predicting outcomes, finding patterns and cycles which are essential in cryptography, computer algorithms, and advanced mathematics.