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
?

Question

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
?

BlackTom AI · Accepted Answer

We start by restating the problem in our own terms: In the quotient ring Z/5Z, the coset α is defined as 5Z + 1, which corresponds to the residue class of 1 modulo 5. To find the coset that equals −α, we need the residue class of the additive inverse of 1 in Z/5Z.

Step 1: Determine α modulo 5. Since α = 5Z + 1, its representative modulo 5 is 1, so α corresponds to the class [1] in Z/5Z.

Step 2: Compute the additive inverse of α. The additive inverse of [1] is [−1], which in modulo 5 arithmetic is [4], because −1 ≡ 4 (mod 5). Thus −α corresponds to the coset [4] in Z/5Z.

Step 3: express [4] as a coset of the form 5Z + k. Any integer congruent to 4 modulo 5 can be written as 5Z + 4. However, sometimes people present alternative representatives that differ by multiples of 5, such as 9, 14, etc., since 9 ≡ 4 (mod 5).

Step 4: verify the equivalence with the given format. The coset 5Z + 9 also represents the same class as 5Z + 4 because 9 ≡ 4 (mod 5). Therefore, 5Z + 9 is a valid way to denote −α in Z/5Z.

Note on the provided options: The prompt you supplied contains no answer choices to analyze individually. If options were given, I would map each to a specific representative modulo 5, determine whether it equals −α, and explain why it is correct or incorrect. In this case, the correct description of −α is the coset corresponding to 4 modulo 5, which can be written as 5Z + 4 or equivalently as 5Z + 9, among other equivalent representatives.

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

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 ?

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