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2. For each of the following items, you must perform all of your calculations by hand, showing all of your work. (This means you must exploit modular arithmetic in your computation; each step must be genuinely reasonable to do without a computer.) (a) Compute the least nonnegative integer in the congruence class 5¹8+7Z. (b) Compute the least nonnegative integer in the congruence class 522 + 112.

2. For each of the following items, you must perform all of your calculations by hand, showing all of your work. (This means you must exploit modular arithmetic in your computation; each step must be genuinely reasonable to do without a computer.) (a) Compute the least nonnegative integer in the congruence class 5¹8+7Z. (b) Compute the least nonnegative integer in the congruence class 522 + 112.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Problem Statement For each of the following items, you must perform all of your calculations by hand, showing all of your work. (This means you must exploit modular arithmetic in your computation; each step must be genuinely reasonable to do without a computer.) 1. **Compute the least nonnegative integer in the congruence class \(5^{18} + 7\mathbb{Z}\).** 2. **Compute the least nonnegative integer in the congruence class \(5^{22} + 11\mathbb{Z}\).** ### Detailed Explanation To solve these problems, you should be familiar with modular arithmetic and the properties of congruences. 1. **For \(5^{18} + 7\mathbb{Z}\):** - **Step 1:** Compute \(5^{18} \mod 7\). - **Step 2:** Show all intermediate steps, using properties such as Fermat's Little Theorem or successive squaring. - **Step 3:** Simplify the result to find the smallest nonnegative integer that is congruent to \(5^{18} \mod 7\). 2. **For \(5^{22} + 11\mathbb{Z}\):** - **Step 1:** Compute \(5^{22} \mod 11\). - **Step 2:** Show all intermediate steps, leveraging appropriate modular arithmetic principles. - **Step 3:** Reduce the result to determine the least nonnegative integer equivalent to \(5^{22} \mod 11\). ### Supporting Diagrams or Graphs This problem involves mathematical calculations rather than visual data; therefore, no specific graphs or diagrams are included. However, step-by-step calculations should be clearly shown in your solution to make the process understandable and verifiable without the use of a computer. ### Educational Context This exercise is geared towards students who are learning about number theory and modular arithmetic. It encourages them to practice manual computation and develop a deeper understanding of congruence relations, exponentiation in modular systems, and the application of theorems in number theory.
Transcribed Image Text:### Problem Statement For each of the following items, you must perform all of your calculations by hand, showing all of your work. (This means you must exploit modular arithmetic in your computation; each step must be genuinely reasonable to do without a computer.) 1. **Compute the least nonnegative integer in the congruence class \(5^{18} + 7\mathbb{Z}\).** 2. **Compute the least nonnegative integer in the congruence class \(5^{22} + 11\mathbb{Z}\).** ### Detailed Explanation To solve these problems, you should be familiar with modular arithmetic and the properties of congruences. 1. **For \(5^{18} + 7\mathbb{Z}\):** - **Step 1:** Compute \(5^{18} \mod 7\). - **Step 2:** Show all intermediate steps, using properties such as Fermat's Little Theorem or successive squaring. - **Step 3:** Simplify the result to find the smallest nonnegative integer that is congruent to \(5^{18} \mod 7\). 2. **For \(5^{22} + 11\mathbb{Z}\):** - **Step 1:** Compute \(5^{22} \mod 11\). - **Step 2:** Show all intermediate steps, leveraging appropriate modular arithmetic principles. - **Step 3:** Reduce the result to determine the least nonnegative integer equivalent to \(5^{22} \mod 11\). ### Supporting Diagrams or Graphs This problem involves mathematical calculations rather than visual data; therefore, no specific graphs or diagrams are included. However, step-by-step calculations should be clearly shown in your solution to make the process understandable and verifiable without the use of a computer. ### Educational Context This exercise is geared towards students who are learning about number theory and modular arithmetic. It encourages them to practice manual computation and develop a deeper understanding of congruence relations, exponentiation in modular systems, and the application of theorems in number theory.
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