Consider the following equation in
𝑍
16
:

8
𝑥
=
0

How many solutions exist for
𝑥
?

Question

Consider the following equation in 
𝑍
16
:

8
𝑥
=
0

How many solutions exist for 
𝑥
?

BlackTom AI · Accepted Answer

We are given the congruence 8x ≡ 0 (mod 16) and asked how many solutions x exist in Z16.
First, interpret the condition 16 | 8x. Since 16 = 2^4 and 8 = 2^3, we have 8x is divisible by 16 exactly when x supplies at least one more factor of 2, i.e., when x is even. In other words, x must be a multiple of 2.
Within the ring Z16, the elements are 0, 1, 2, ..., 15. The even residues among these are 0, 2, 4, 6, 8, 10, 12, 14. There are 8 such elements, each satisfying 8x ≡ 0 (mod 16).
Thus, the equation has eight distinct solutions in Z16. Because the answer options are missing in the provided data, I’m focusing on the logical deduction: the condition reduces to x being even modulo 16, and there are exactly eight even residues modulo 16. If a multiple-choice list were present, the correct choice would correspond to describing eight solutions, i.e., the eight even residues in Z16.

MATH3066 (ND) 2025 Mid-Semester Test Practice Set C

Consider the following equation in 𝑍 16 : 8 𝑥 = 0 How many solutions exist for 𝑥 ?

View Explanation

Log in for full answers

Similar Questions

Solve

For which of the following would modular division be useful? 下列哪项对于模除有用？

Consider the field 𝑍 / 5 𝑍 and the coset 𝛼 = 5 𝑍 + 2 . Which one of the following cosets is equal to 𝛼 − 1 ?

Consider the quotient ring 𝑍 / 5 𝑍 and the coset 𝛼 = 5 𝑍 + 1 . Which one of the following cosets is equal to − 𝛼 , the negative of 𝛼 ?

Consider the following equation in 𝑍 27 : 3 𝑥 = 7 How many solutions exist for 𝑥 ?

Simplify the following expression, working over 𝑍 13 : 3 − 1 2 10 =

Simplify the following expression, working over :

Suppose n\mod d = 3. Which of the following is not necessarily true?

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler