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(b) It remains to show that U(n) is closed under inverse. Suppose that m € U(n) and a is the inverse of m. What modular equation must satisfy? (*Hint*) (c) Show that the equation in a that you wrote in part (b) has a solution as long as m is relatively prime to n.

(b) It remains to show that U(n) is closed under inverse. Suppose that m € U(n) and a is the inverse of m. What modular equation must satisfy? (*Hint*) (c) Show that the equation in a that you wrote in part (b) has a solution as long as m is relatively prime to n.

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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Please do part B and C and please show step by step and explain

Exercise 15.2.26. In this exercise, we prove that \( U(n) \) is a group under multiplication mod \( n \) for any \( n \). We know that modular multiplication is associative, so it remains to show the closure and inverse properties. (a) Fill in the blanks to show that \( U(n) \) is closed under modular multiplication: Let \( k, m \) be arbitrary elements of \( U(n) \). It follows that both \( k \) and \( \langle 1 \rangle \) are relatively prime to \( \langle 2 \rangle \). So neither \( k \) nor \( \langle 3 \rangle \) has any prime factors in common with \( \langle 4 \rangle \). It follows that the product \( \langle 5 \rangle \) also has no prime factors in common with \( \langle 6 \rangle \). Furthermore, the remainder of \( \langle 7 \rangle \) under division by \( \langle 8 \rangle \) also has no prime factors in common with \( \langle 9 \rangle \). Therefore the product of \( \langle 10 \rangle \) and \( \langle 11 \rangle \) under modular multiplication is also an element of \( \langle 12 \rangle \), so \( \langle 13 \rangle \) is closed under modular multiplication. (b) It remains to show that \( U(n) \) is closed under inverse. Suppose that \( m \in U(n) \) and \( x \) is the inverse of \( m \). What modular equation must \( x \) satisfy? (*Hint*) (c) Show that the equation in \( x \) that you wrote in part (b) has a solution as long as \( m \) is relatively prime to \( n \).
Transcribed Image Text:Exercise 15.2.26. In this exercise, we prove that \( U(n) \) is a group under multiplication mod \( n \) for any \( n \). We know that modular multiplication is associative, so it remains to show the closure and inverse properties. (a) Fill in the blanks to show that \( U(n) \) is closed under modular multiplication: Let \( k, m \) be arbitrary elements of \( U(n) \). It follows that both \( k \) and \( \langle 1 \rangle \) are relatively prime to \( \langle 2 \rangle \). So neither \( k \) nor \( \langle 3 \rangle \) has any prime factors in common with \( \langle 4 \rangle \). It follows that the product \( \langle 5 \rangle \) also has no prime factors in common with \( \langle 6 \rangle \). Furthermore, the remainder of \( \langle 7 \rangle \) under division by \( \langle 8 \rangle \) also has no prime factors in common with \( \langle 9 \rangle \). Therefore the product of \( \langle 10 \rangle \) and \( \langle 11 \rangle \) under modular multiplication is also an element of \( \langle 12 \rangle \), so \( \langle 13 \rangle \) is closed under modular multiplication. (b) It remains to show that \( U(n) \) is closed under inverse. Suppose that \( m \in U(n) \) and \( x \) is the inverse of \( m \). What modular equation must \( x \) satisfy? (*Hint*) (c) Show that the equation in \( x \) that you wrote in part (b) has a solution as long as \( m \) is relatively prime to \( n \).
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