6. Prove o (k2) = k · Þ(k) for all k E N.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question

6

1. Use Euler's Theorem to prove
a265 = a (mod 105)
for all a E Z.
2. Use Fermat's Little Theorem, or its corollary, to find the units digit of
72018 + 112019 +
+ 132020 + 132021 + 172022
3. Use Wilson's Theorem to prove 6(k – 4)! = 1 (mod k), if k is prime.
4. Use Fermat's factorization method to factor 2168495737.
5. Use Kraitchik's factorization method to factor 11653.
6. Prove o (k2) = k · 4(k) for all k E N.
%3D
7. Prove each of the following statements.
(a) If q is a prime number not equal to 3 and k = 3q, then o(k) = 2 (T(k) + ¢(k)).
%3|
%3D
(b) If q is an odd prime number and k = 2q, then k = o(k) – T(k) – (k).
%3D
8. Let â be the inverse of a modulo k. Prove that the order of a modulo k is equal to the order of â modulo
k. Use this result to easily show that if a is a primitive root modulo k then â is also a primitive root
modulo k.
Transcribed Image Text:1. Use Euler's Theorem to prove a265 = a (mod 105) for all a E Z. 2. Use Fermat's Little Theorem, or its corollary, to find the units digit of 72018 + 112019 + + 132020 + 132021 + 172022 3. Use Wilson's Theorem to prove 6(k – 4)! = 1 (mod k), if k is prime. 4. Use Fermat's factorization method to factor 2168495737. 5. Use Kraitchik's factorization method to factor 11653. 6. Prove o (k2) = k · 4(k) for all k E N. %3D 7. Prove each of the following statements. (a) If q is a prime number not equal to 3 and k = 3q, then o(k) = 2 (T(k) + ¢(k)). %3| %3D (b) If q is an odd prime number and k = 2q, then k = o(k) – T(k) – (k). %3D 8. Let â be the inverse of a modulo k. Prove that the order of a modulo k is equal to the order of â modulo k. Use this result to easily show that if a is a primitive root modulo k then â is also a primitive root modulo k.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 2 images

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,