**Title: Computing the Orders of Sets in Modular Arithmetic****Objective:** Learn how to compute the orders of the sets $ U(5) $, $ U(7) $, and $ U(35) $.---**Introduction to Modular Arithmetic:**In number theory, the set $ U(n) $ represents the group of integers less than $ n $ that are coprime with $ n $. The "order" of $ U(n) $ is the number of elements in this set, often calculated using Euler's totient function $ \phi(n) $.**Steps to Calculate the Orders:**1. **Understanding $ U(5) $:** - Euler's totient function $ \phi(n) $ gives the count of integers up to $ n $ that are coprime with $ n $. - For a prime number $ p $, $ \phi(p) = p - 1 $. - Therefore, $ \phi(5) = 4 $.2. **Calculating $ U(7) $:** - Here again, $ 7 $ is a prime number. So, $ \phi(7) = 7 - 1 = 6 $.3. **Finding $ U(35) $:** - If $ n $ is a product of two coprime integers $ a $ and $ b $, $ \phi(n) = \phi(a) \times \phi(b) $. - Since $ 35 = 5 \times 7 $ (where both 5 and 7 are primes), $ \phi(35) = \phi(5) \times \phi(7) = 4 \times 6 = 24 $.**Conclusion:**- The orders of the sets are $ U(5) = 4 $, $ U(7) = 6 $, and $ U(35) = 24 $.- Understanding these fundamental theorems and functions in number theory helps in various cryptographic algorithms.---This article provides a concise introduction to computing orders using foundational concepts in number theory.

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Description: **Title: Computing the Orders of Sets in Modular Arithmetic****Objective:** Learn how to compute the orders of the sets $ U(5) $, $ U(7) $, and $ U(35) $.---**Introduction to Modular Arithmetic:**In number theory, the set $ U(n) $ represent

Note that, U(n) represents a group consisting of non-negative integers which are less than n and are…

Answered: Compute the orders of: U(5), U(7),…

Math

Advanced Math

Compute the orders of: U(5), U(7), U(35)

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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**Title: Computing the Orders of Sets in Modular Arithmetic**

**Objective:**
Learn how to compute the orders of the sets $ U(5) $, $ U(7) $, and $ U(35) $.

---

**Introduction to Modular Arithmetic:**

In number theory, the set $ U(n) $ represents the group of integers less than $ n $ that are coprime with $ n $. The "order" of $ U(n) $ is the number of elements in this set, often calculated using Euler's totient function $ \phi(n) $.

**Steps to Calculate the Orders:**

1. **Understanding $ U(5) $:**
- Euler's totient function $ \phi(n) $ gives the count of integers up to $ n $ that are coprime with $ n $.
- For a prime number $ p $, $ \phi(p) = p - 1 $.
- Therefore, $ \phi(5) = 4 $.

2. **Calculating $ U(7) $:**
- Here again, $ 7 $ is a prime number. So, $ \phi(7) = 7 - 1 = 6 $.

3. **Finding $ U(35) $:**
- If $ n $ is a product of two coprime integers $ a $ and $ b $, $ \phi(n) = \phi(a) \times \phi(b) $.
- Since $ 35 = 5 \times 7 $ (where both 5 and 7 are primes), $ \phi(35) = \phi(5) \times \phi(7) = 4 \times 6 = 24 $.

**Conclusion:**

- The orders of the sets are $ U(5) = 4 $, $ U(7) = 6 $, and $ U(35) = 24 $.
- Understanding these fundamental theorems and functions in number theory helps in various cryptographic algorithms.

---

This article provides a concise introduction to computing orders using foundational concepts in number theory.

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