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 ?

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