Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Exercise 18.3.13. (a) Verify Euler's theorem for n = 15 and a = 4. (b) Verify Euler's theorem for n = 22 and a = 3.

Exercise 18.3.13. (a) Verify Euler's theorem for n = 15 and a = 4. (b) Verify Euler's theorem for n = 22 and a = 3.

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
See similar textbooks
icon
Related questions
Question

Please do Exercise 18.3.13 part A and B and please show step by step and explain

**Proposition 18.3.12. (Euler’s theorem)** Let \( a \) and \( n \) be integers such that \( n > 0 \) and \(\gcd(a, n) = 1\). Then \( a^{\phi(n)} \equiv 1 \pmod{n} \). **Proof.** First, let \( r \) be the remainder when \( a \) is divided by \( n \). We may consider \( r \) as an element of \( U(n) \). As noted above, the order of \( U(n) \) is \( \phi(n) \). Lagrange’s theorem then tells us that \(|r| \) divides \(\phi(n)\), so we can write: \(\phi(n) = k|r|\), where \( k \in \mathbb{N} \). Consequently, considering \( r \) as an element of \( U(n) \), we have \( r^{\phi(n)} = r^{k|r|} = (r^{|r|})^k = (1)^k = 1 \) (take note that the multiplication that is being used here is modular multiplication, not regular multiplication). Finally, we may use the fact that \( a \equiv r \pmod{n} \) and apply Exercise 5.4.7 in Section 5.4.1 to conclude that \( a^{\phi(n)} \equiv 1 \pmod{n} \). **Exercise 18.3.13.** (a) Verify Euler’s theorem for \( n = 15 \) and \( a = 4 \). (b) Verify Euler’s theorem for \( n = 22 \) and \( a = 3 \).
Transcribed Image Text:**Proposition 18.3.12. (Euler’s theorem)** Let \( a \) and \( n \) be integers such that \( n > 0 \) and \(\gcd(a, n) = 1\). Then \( a^{\phi(n)} \equiv 1 \pmod{n} \). **Proof.** First, let \( r \) be the remainder when \( a \) is divided by \( n \). We may consider \( r \) as an element of \( U(n) \). As noted above, the order of \( U(n) \) is \( \phi(n) \). Lagrange’s theorem then tells us that \(|r| \) divides \(\phi(n)\), so we can write: \(\phi(n) = k|r|\), where \( k \in \mathbb{N} \). Consequently, considering \( r \) as an element of \( U(n) \), we have \( r^{\phi(n)} = r^{k|r|} = (r^{|r|})^k = (1)^k = 1 \) (take note that the multiplication that is being used here is modular multiplication, not regular multiplication). Finally, we may use the fact that \( a \equiv r \pmod{n} \) and apply Exercise 5.4.7 in Section 5.4.1 to conclude that \( a^{\phi(n)} \equiv 1 \pmod{n} \). **Exercise 18.3.13.** (a) Verify Euler’s theorem for \( n = 15 \) and \( a = 4 \). (b) Verify Euler’s theorem for \( n = 22 \) and \( a = 3 \).
Expert Solution
steps

Step by step

Solved in 2 steps with 1 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS