Question

(a) Find all solutions of the equation $a^2 \equiv 1 \pmod{8}$, for $a \in \{1, 2, ..., 7\}$. (b) Let $p$ be prime and $a \in \{1,2,..., p-1\}$. Determine whether the following statement is true or false, and provide a justification for your answer: If $a^3 \equiv 1 \pmod{p}$ then $a \equiv 1 \pmod{p}$.

          (a) Find all solutions of the equation $a^2 \equiv 1 \pmod{8}$, for $a \in \{1, 2, ..., 7\}$.
(b) Let $p$ be prime and $a \in \{1,2,..., p-1\}$. Determine whether the following statement is
true or false, and provide a justification for your answer:
If $a^3 \equiv 1 \pmod{p}$ then $a \equiv 1 \pmod{p}$.

$(a) Find all solutions of the equation a^2 ≡ 1 8, for a ∈{1, 2, ..., 7}. (b) Let p be prime and a ∈{1,2,..., p-1}. Determine whether the following statement is true or false, and provide a justification for your answer: If a^3 ≡ 1 p then a ≡ 1 p.$

Added by Jose C.

Computer Science and Information Technology

Trishna Knowledge Systems 2018 Edition

Instant Answer

Solved by Expert Shaiju T

06/07/2022

Step 1

(a) There are six solutions: 1, 1, 3, 5, 7, 11. Show more…

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(a) Find all solutions of the equation a^2 â‰¡ 1 (mod 8), for a âˆˆ {1,2,...,7}. (b) Let p be a prime number and a âˆˆ {1,2,...,p-1}. Determine whether the following statement is true or false, and provide a justification for your answer: If a^3 â‰¡ 1 (mod p), then a â‰¡ 1 (mod p).