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What is the order of 3 modulo 101? Is 3 a primitive root modulo 101?

What is the order of 3 modulo 101? Is 3 a primitive root modulo 101?

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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I'm sorry, I can't transcribe the part covered in blue. However, based on the visible text: **Educational Exercise: Understanding Modulo and Primitive Roots** To explore concepts in modular arithmetic, consider the following problem: (7) What is the order of 3 modulo 101? Is 3 a primitive root modulo 101? ### Instructions: - First, calculate the order of 3 modulo 101, which is the smallest positive integer \( k \) such that \( 3^k \equiv 1 \pmod{101} \). - Then, determine if 3 is a primitive root modulo 101 by checking whether 3 generates all non-zero residues modulo 101. This exercise encourages the understanding of cyclic groups and the application of Fermat's Little Theorem in number theory.
Transcribed Image Text:I'm sorry, I can't transcribe the part covered in blue. However, based on the visible text: **Educational Exercise: Understanding Modulo and Primitive Roots** To explore concepts in modular arithmetic, consider the following problem: (7) What is the order of 3 modulo 101? Is 3 a primitive root modulo 101? ### Instructions: - First, calculate the order of 3 modulo 101, which is the smallest positive integer \( k \) such that \( 3^k \equiv 1 \pmod{101} \). - Then, determine if 3 is a primitive root modulo 101 by checking whether 3 generates all non-zero residues modulo 101. This exercise encourages the understanding of cyclic groups and the application of Fermat's Little Theorem in number theory.
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