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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
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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 ged(a, n) = 1. Then a(n) = 1 (mod 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 o(n). Lagrange's theorem then tells us that |r| divides o(n), so we can write: (n) = kr, where k € N. Consequently, considering r as an element of U(n), we have r(n) = || = (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 = r (mod n) and apply Exercise 5.4.7 in Section 5.4.1 to conclude that a(n) = 1 (mod 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 ged(a, n) = 1. Then a(n) = 1 (mod 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 o(n). Lagrange's theorem then tells us that |r| divides o(n), so we can write: (n) = kr, where k € N. Consequently, considering r as an element of U(n), we have r(n) = || = (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 = r (mod n) and apply Exercise 5.4.7 in Section 5.4.1 to conclude that a(n) = 1 (mod 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.
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