Consider the field
𝑍
/
5
𝑍
and the coset

𝛼
=
5
𝑍
+
2
.

Which one of the following cosets is equal to
𝛼
−
1
?

Question

Consider the field 
𝑍
/
5
𝑍
 and the coset

𝛼
=
5
𝑍
+
2
.

Which one of the following cosets is equal to 
𝛼
−
1
?

BlackTom AI · Accepted Answer

Let's unpack the problem step by step and keep track of what each symbol means in the quotient Z/5Z.
First, the coset α is given as 5Z + 2, which represents the residue class of 2 modulo 5. In other words, α corresponds to the equivalence class [2] in Z/5Z.
Now, to find α − 1, we subtract 1 from the representative inside the same coset structure: α − 1 = (5Z + 2) − 1 = 5Z + 1. This is the coset corresponding to the residue class [1] modulo 5.
Next, examine the option provided: 5Z + 13. Since 13 ≡ 3 (mod 5), the coset 5Z + 13 represents the residue class [3] in Z/5Z. This is not the same as [1], so this option is not equal to α − 1.
To see the discrepancy clearly, convert all representatives modulo 5:
- α = 5Z + 2 → residue 2
- α − 1 = 5Z + 1 → residue 1
- 5Z + 13 → residue 3
Thus, the given option corresponds to residue 3, which does not match the required residue 1 for α − 1.
Note: The problem statement provides only one option to evaluate. Since there are no other answer choices listed, we cannot compare multiple alternatives here. What we can assert is that the supplied option 5Z + 13 does not equal α − 1, given α = 5Z + 2.
If additional options were present, we would similarly translate each to its residue class modulo 5 and compare to 5Z + 1 to determine which coset matches α − 1.

MATH3066 (ND) 2025 MATH3066 Exam Practice Multiple Choice Questions D

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

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