Assignment 4

Question 1

Have you read Sections 2.1 and 2.2? Circle one: Yes / No.

Question 2

Consider the congruence class \([2] \in \mathbb{Z}_5\). List 6 elements of the set \([2]\).

Question 3

How many elements are in \(\mathbb{Z}_4\)? List them.

\(|\mathbb{Z}_4| = \)
\(\mathbb{Z}_4 = \)

Question 4

Consider the congruence class \([33]\) of \(\mathbb{Z}_7\); find the smallest positive integer that is a representative of \([33]\).

Question 5

Find the smallest positive integer \(b\) that is a solution of the congruence \(9b \equiv 1 \pmod{11}\).

Question 6

Prove that if \((a, n) = 1\) then there exists an integer \(b\) such that \(ab \equiv 1 \pmod{n}\).

Question 7

Give a specific counterexample to disprove: If \( ab \equiv 0 \pmod{n} \) then \( a \equiv 0 \pmod{n} \) or \( b \equiv 0 \pmod{n} \).

Question 8

Prove the transitivity of congruence: If \( a \equiv b \pmod{n} \) and \( b \equiv c \pmod{n} \) then \( a \equiv c \pmod{n} \).

Question 9

Find all integers \(x\) such that \( x^2 - 7x + 12 \equiv 0 \pmod{7} \). Prove that your answer is correct.