Proper-Ideal Cosets and Representative-Independent Quotient Operations
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Build the Quotient from Full Additive Cosets
Use the parent ring Z_4 with residues 0 through 3 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).
View answer and explanation
([0, 2], [[0, 2], [1, 3]], 2, proper_two_sided_ideal_in_commutative_Zn)
List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 2], [[0, 2], [1, 3]], 2, proper_two_sided_ideal_in_commutative_Zn).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate
2. Build the Quotient from Full Additive Cosets
Use the parent ring Z_6 with residues 0 through 5 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).
View answer and explanation
([0, 3], [[0, 3], [1, 4], [2, 5]], 3, proper_two_sided_ideal_in_commutative_Zn)
List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 3], [[0, 3], [1, 4], [2, 5]], 3, proper_two_sided_ideal_in_commutative_Zn).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate
3. Build the Quotient from Full Additive Cosets
Use the parent ring Z_8 with residues 0 through 7 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).
View answer and explanation
([0, 4], [[0, 4], [1, 5], [2, 6], [3, 7]], 4, proper_two_sided_ideal_in_commutative_Zn)
List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 4], [[0, 4], [1, 5], [2, 6], [3, 7]], 4, proper_two_sided_ideal_in_commutative_Zn).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate
4. Build the Quotient from Full Additive Cosets
Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).
View answer and explanation
([0, 5], [[0, 5], [1, 6], [2, 7], [3, 8], [4, 9]], 5, proper_two_sided_ideal_in_commutative_Zn)
List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 5], [[0, 5], [1, 6], [2, 7], [3, 8], [4, 9]], 5, proper_two_sided_ideal_in_commutative_Zn).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate
5. Build the Quotient from Full Additive Cosets
Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).
View answer and explanation
([0, 6], [[0, 6], [1, 7], [2, 8], [3, 9], [4, 10], [5, 11]], 6, proper_two_sided_ideal_in_commutative_Zn)
List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 6], [[0, 6], [1, 7], [2, 8], [3, 9], [4, 10], [5, 11]], 6, proper_two_sided_ideal_in_commutative_Zn).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate
6. Build the Quotient from Full Additive Cosets
Use the parent ring Z_6 with residues 0 through 5 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).
View answer and explanation
([0, 2, 4], [[0, 2, 4], [1, 3, 5]], 2, proper_two_sided_ideal_in_commutative_Zn)
List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 2, 4], [[0, 2, 4], [1, 3, 5]], 2, proper_two_sided_ideal_in_commutative_Zn).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate
7. Build the Quotient from Full Additive Cosets
Use the parent ring Z_9 with residues 0 through 8 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).
View answer and explanation
([0, 3, 6], [[0, 3, 6], [1, 4, 7], [2, 5, 8]], 3, proper_two_sided_ideal_in_commutative_Zn)
List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 3, 6], [[0, 3, 6], [1, 4, 7], [2, 5, 8]], 3, proper_two_sided_ideal_in_commutative_Zn).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate
8. Build the Quotient from Full Additive Cosets
Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).
View answer and explanation
([0, 4, 8], [[0, 4, 8], [1, 5, 9], [2, 6, 10], [3, 7, 11]], 4, proper_two_sided_ideal_in_commutative_Zn)
List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 4, 8], [[0, 4, 8], [1, 5, 9], [2, 6, 10], [3, 7, 11]], 4, proper_two_sided_ideal_in_commutative_Zn).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate
9. Build the Quotient from Full Additive Cosets
Use the parent ring Z_15 with residues 0 through 14 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).
View answer and explanation
([0, 5, 10], [[0, 5, 10], [1, 6, 11], [2, 7, 12], [3, 8, 13], [4, 9, 14]], 5, proper_two_sided_ideal_in_commutative_Zn)
List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 5, 10], [[0, 5, 10], [1, 6, 11], [2, 7, 12], [3, 8, 13], [4, 9, 14]], 5, proper_two_sided_ideal_in_commutative_Zn).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate
10. Build the Quotient from Full Additive Cosets
Use the parent ring Z_18 with residues 0 through 17 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).
View answer and explanation
([0, 6, 12], [[0, 6, 12], [1, 7, 13], [2, 8, 14], [3, 9, 15], [4, 10, 16], [5, 11, 17]], 6, proper_two_sided_ideal_in_commutative_Zn)
List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 6, 12], [[0, 6, 12], [1, 7, 13], [2, 8, 14], [3, 9, 15], [4, 10, 16], [5, 11, 17]], 6, proper_two_sided_ideal_in_commutative_Zn).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate
11. Build the Quotient from Full Additive Cosets
Use the parent ring Z_8 with residues 0 through 7 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).
View answer and explanation
([0, 2, 4, 6], [[0, 2, 4, 6], [1, 3, 5, 7]], 2, proper_two_sided_ideal_in_commutative_Zn)
List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 2, 4, 6], [[0, 2, 4, 6], [1, 3, 5, 7]], 2, proper_two_sided_ideal_in_commutative_Zn).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate
12. Build the Quotient from Full Additive Cosets
Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).
View answer and explanation
([0, 3, 6, 9], [[0, 3, 6, 9], [1, 4, 7, 10], [2, 5, 8, 11]], 3, proper_two_sided_ideal_in_commutative_Zn)
List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 3, 6, 9], [[0, 3, 6, 9], [1, 4, 7, 10], [2, 5, 8, 11]], 3, proper_two_sided_ideal_in_commutative_Zn).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate
13. Build the Quotient from Full Additive Cosets
Use the parent ring Z_16 with residues 0 through 15 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).
View answer and explanation
([0, 4, 8, 12], [[0, 4, 8, 12], [1, 5, 9, 13], [2, 6, 10, 14], [3, 7, 11, 15]], 4, proper_two_sided_ideal_in_commutative_Zn)
List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 4, 8, 12], [[0, 4, 8, 12], [1, 5, 9, 13], [2, 6, 10, 14], [3, 7, 11, 15]], 4, proper_two_sided_ideal_in_commutative_Zn).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate
14. Build the Quotient from Full Additive Cosets
Use the parent ring Z_20 with residues 0 through 19 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).
View answer and explanation
([0, 5, 10, 15], [[0, 5, 10, 15], [1, 6, 11, 16], [2, 7, 12, 17], [3, 8, 13, 18], [4, 9, 14, 19]], 5, proper_two_sided_ideal_in_commutative_Zn)
List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 5, 10, 15], [[0, 5, 10, 15], [1, 6, 11, 16], [2, 7, 12, 17], [3, 8, 13, 18], [4, 9, 14, 19]], 5, proper_two_sided_ideal_in_commutative_Zn).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate
15. Build the Quotient from Full Additive Cosets
Use the parent ring Z_24 with residues 0 through 23 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).
View answer and explanation
([0, 6, 12, 18], [[0, 6, 12, 18], [1, 7, 13, 19], [2, 8, 14, 20], [3, 9, 15, 21], [4, 10, 16, 22], [5, 11, 17, 23]], 6, proper_two_sided_ideal_in_commutative_Zn)
List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 6, 12, 18], [[0, 6, 12, 18], [1, 7, 13, 19], [2, 8, 14, 20], [3, 9, 15, 21], [4, 10, 16, 22], [5, 11, 17, 23]], 6, proper_two_sided_ideal_in_commutative_Zn).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate
16. Build the Quotient from Full Additive Cosets
Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).
View answer and explanation
([0, 2, 4, 6, 8], [[0, 2, 4, 6, 8], [1, 3, 5, 7, 9]], 2, proper_two_sided_ideal_in_commutative_Zn)
List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8], [[0, 2, 4, 6, 8], [1, 3, 5, 7, 9]], 2, proper_two_sided_ideal_in_commutative_Zn).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate
17. Build the Quotient from Full Additive Cosets
Use the parent ring Z_15 with residues 0 through 14 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).
View answer and explanation
([0, 3, 6, 9, 12], [[0, 3, 6, 9, 12], [1, 4, 7, 10, 13], [2, 5, 8, 11, 14]], 3, proper_two_sided_ideal_in_commutative_Zn)
List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 3, 6, 9, 12], [[0, 3, 6, 9, 12], [1, 4, 7, 10, 13], [2, 5, 8, 11, 14]], 3, proper_two_sided_ideal_in_commutative_Zn).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate
18. Build the Quotient from Full Additive Cosets
Use the parent ring Z_20 with residues 0 through 19 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).
View answer and explanation
([0, 4, 8, 12, 16], [[0, 4, 8, 12, 16], [1, 5, 9, 13, 17], [2, 6, 10, 14, 18], [3, 7, 11, 15, 19]], 4, proper_two_sided_ideal_in_commutative_Zn)
List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 4, 8, 12, 16], [[0, 4, 8, 12, 16], [1, 5, 9, 13, 17], [2, 6, 10, 14, 18], [3, 7, 11, 15, 19]], 4, proper_two_sided_ideal_in_commutative_Zn).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate
19. Build the Quotient from Full Additive Cosets
Use the parent ring Z_25 with residues 0 through 24 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).
View answer and explanation
([0, 5, 10, 15, 20], [[0, 5, 10, 15, 20], [1, 6, 11, 16, 21], [2, 7, 12, 17, 22], [3, 8, 13, 18, 23], [4, 9, 14, 19, 24]], 5, proper_two_sided_ideal_in_commutative_Zn)
List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 5, 10, 15, 20], [[0, 5, 10, 15, 20], [1, 6, 11, 16, 21], [2, 7, 12, 17, 22], [3, 8, 13, 18, 23], [4, 9, 14, 19, 24]], 5, proper_two_sided_ideal_in_commutative_Zn).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate
20. Build the Quotient from Full Additive Cosets
Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).
View answer and explanation
([0, 6, 12, 18, 24], [[0, 6, 12, 18, 24], [1, 7, 13, 19, 25], [2, 8, 14, 20, 26], [3, 9, 15, 21, 27], [4, 10, 16, 22, 28], [5, 11, 17, 23, 29]], 6, proper_two_sided_ideal_in_commutative_Zn)
List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 6, 12, 18, 24], [[0, 6, 12, 18, 24], [1, 7, 13, 19, 25], [2, 8, 14, 20, 26], [3, 9, 15, 21, 27], [4, 10, 16, 22, 28], [5, 11, 17, 23, 29]], 6, proper_two_sided_ideal_in_commutative_Zn).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate
21. Build the Quotient from Full Additive Cosets
Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).
View answer and explanation
([0, 2, 4, 6, 8, 10], [[0, 2, 4, 6, 8, 10], [1, 3, 5, 7, 9, 11]], 2, proper_two_sided_ideal_in_commutative_Zn)
List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10], [[0, 2, 4, 6, 8, 10], [1, 3, 5, 7, 9, 11]], 2, proper_two_sided_ideal_in_commutative_Zn).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate
22. Build the Quotient from Full Additive Cosets
Use the parent ring Z_18 with residues 0 through 17 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).
View answer and explanation
([0, 3, 6, 9, 12, 15], [[0, 3, 6, 9, 12, 15], [1, 4, 7, 10, 13, 16], [2, 5, 8, 11, 14, 17]], 3, proper_two_sided_ideal_in_commutative_Zn)
List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 3, 6, 9, 12, 15], [[0, 3, 6, 9, 12, 15], [1, 4, 7, 10, 13, 16], [2, 5, 8, 11, 14, 17]], 3, proper_two_sided_ideal_in_commutative_Zn).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate
23. Build the Quotient from Full Additive Cosets
Use the parent ring Z_24 with residues 0 through 23 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).
View answer and explanation
([0, 4, 8, 12, 16, 20], [[0, 4, 8, 12, 16, 20], [1, 5, 9, 13, 17, 21], [2, 6, 10, 14, 18, 22], [3, 7, 11, 15, 19, 23]], 4, proper_two_sided_ideal_in_commutative_Zn)
List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 4, 8, 12, 16, 20], [[0, 4, 8, 12, 16, 20], [1, 5, 9, 13, 17, 21], [2, 6, 10, 14, 18, 22], [3, 7, 11, 15, 19, 23]], 4, proper_two_sided_ideal_in_commutative_Zn).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate
24. Build the Quotient from Full Additive Cosets
Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).
View answer and explanation
([0, 5, 10, 15, 20, 25], [[0, 5, 10, 15, 20, 25], [1, 6, 11, 16, 21, 26], [2, 7, 12, 17, 22, 27], [3, 8, 13, 18, 23, 28], [4, 9, 14, 19, 24, 29]], 5, proper_two_sided_ideal_in_commutative_Zn)
List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 5, 10, 15, 20, 25], [[0, 5, 10, 15, 20, 25], [1, 6, 11, 16, 21, 26], [2, 7, 12, 17, 22, 27], [3, 8, 13, 18, 23, 28], [4, 9, 14, 19, 24, 29]], 5, proper_two_sided_ideal_in_commutative_Zn).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate
25. Build the Quotient from Full Additive Cosets
Use the parent ring Z_36 with residues 0 through 35 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).
View answer and explanation
([0, 6, 12, 18, 24, 30], [[0, 6, 12, 18, 24, 30], [1, 7, 13, 19, 25, 31], [2, 8, 14, 20, 26, 32], [3, 9, 15, 21, 27, 33], [4, 10, 16, 22, 28, 34], [5, 11, 17, 23, 29, 35]], 6, proper_two_sided_ideal_in_commutative_Zn)
List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 6, 12, 18, 24, 30], [[0, 6, 12, 18, 24, 30], [1, 7, 13, 19, 25, 31], [2, 8, 14, 20, 26, 32], [3, 9, 15, 21, 27, 33], [4, 10, 16, 22, 28, 34], [5, 11, 17, 23, 29, 35]], 6, proper_two_sided_ideal_in_commutative_Zn).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-classes · candidate
26. Add in the Parent Ring, Then Take the Quotient Class
Use the parent ring Z_4 with residues 0 through 3 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=0, b=1. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).
View answer and explanation
(0, 1, 1, 1, [a]+[b]=[a+b])
Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (0, 1, 1, 1, [a]+[b]=[a+b]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-add · candidate
27. Add in the Parent Ring, Then Take the Quotient Class
Use the parent ring Z_6 with residues 0 through 5 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=1, b=3. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).
View answer and explanation
(1, 0, 4, 1, [a]+[b]=[a+b])
Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (1, 0, 4, 1, [a]+[b]=[a+b]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-add · candidate
28. Add in the Parent Ring, Then Take the Quotient Class
Use the parent ring Z_8 with residues 0 through 7 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=2, b=5. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).
View answer and explanation
(2, 1, 7, 3, [a]+[b]=[a+b])
Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (2, 1, 7, 3, [a]+[b]=[a+b]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-add · candidate
29. Add in the Parent Ring, Then Take the Quotient Class
Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=3, b=7. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).
View answer and explanation
(3, 2, 0, 0, [a]+[b]=[a+b])
Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (3, 2, 0, 0, [a]+[b]=[a+b]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-add · candidate
30. Add in the Parent Ring, Then Take the Quotient Class
Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=4, b=9. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).
View answer and explanation
(4, 3, 1, 1, [a]+[b]=[a+b])
Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (4, 3, 1, 1, [a]+[b]=[a+b]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-add · candidate
31. Add in the Parent Ring, Then Take the Quotient Class
Use the parent ring Z_6 with residues 0 through 5 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=5, b=5. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).
View answer and explanation
(1, 1, 4, 0, [a]+[b]=[a+b])
Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (1, 1, 4, 0, [a]+[b]=[a+b]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-add · candidate
32. Add in the Parent Ring, Then Take the Quotient Class
Use the parent ring Z_9 with residues 0 through 8 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=6, b=4. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).
View answer and explanation
(0, 1, 1, 1, [a]+[b]=[a+b])
Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (0, 1, 1, 1, [a]+[b]=[a+b]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-add · candidate
33. Add in the Parent Ring, Then Take the Quotient Class
Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=7, b=3. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).
View answer and explanation
(3, 3, 10, 2, [a]+[b]=[a+b])
Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (3, 3, 10, 2, [a]+[b]=[a+b]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-add · candidate
34. Add in the Parent Ring, Then Take the Quotient Class
Use the parent ring Z_15 with residues 0 through 14 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=8, b=2. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).
View answer and explanation
(3, 2, 10, 0, [a]+[b]=[a+b])
Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (3, 2, 10, 0, [a]+[b]=[a+b]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-add · candidate
35. Add in the Parent Ring, Then Take the Quotient Class
Use the parent ring Z_18 with residues 0 through 17 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=9, b=1. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).
View answer and explanation
(3, 1, 10, 4, [a]+[b]=[a+b])
Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (3, 1, 10, 4, [a]+[b]=[a+b]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-add · candidate
36. Add in the Parent Ring, Then Take the Quotient Class
Use the parent ring Z_8 with residues 0 through 7 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=2, b=5. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).
View answer and explanation
(0, 1, 7, 1, [a]+[b]=[a+b])
Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (0, 1, 7, 1, [a]+[b]=[a+b]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-add · candidate
37. Add in the Parent Ring, Then Take the Quotient Class
Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=11, b=11. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).
View answer and explanation
(2, 2, 10, 1, [a]+[b]=[a+b])
Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (2, 2, 10, 1, [a]+[b]=[a+b]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-add · candidate
38. Add in the Parent Ring, Then Take the Quotient Class
Use the parent ring Z_16 with residues 0 through 15 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=12, b=9. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).
View answer and explanation
(0, 1, 5, 1, [a]+[b]=[a+b])
Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (0, 1, 5, 1, [a]+[b]=[a+b]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-add · candidate
39. Add in the Parent Ring, Then Take the Quotient Class
Use the parent ring Z_20 with residues 0 through 19 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=13, b=7. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).
View answer and explanation
(3, 2, 0, 0, [a]+[b]=[a+b])
Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (3, 2, 0, 0, [a]+[b]=[a+b]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-add · candidate
40. Add in the Parent Ring, Then Take the Quotient Class
Use the parent ring Z_24 with residues 0 through 23 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=14, b=5. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).
View answer and explanation
(2, 5, 19, 1, [a]+[b]=[a+b])
Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (2, 5, 19, 1, [a]+[b]=[a+b]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-add · candidate
41. Add in the Parent Ring, Then Take the Quotient Class
Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=5, b=1. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).
View answer and explanation
(1, 1, 6, 0, [a]+[b]=[a+b])
Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (1, 1, 6, 0, [a]+[b]=[a+b]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-add · candidate
42. Add in the Parent Ring, Then Take the Quotient Class
Use the parent ring Z_15 with residues 0 through 14 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=1, b=3. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).
View answer and explanation
(1, 0, 4, 1, [a]+[b]=[a+b])
Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (1, 0, 4, 1, [a]+[b]=[a+b]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-add · candidate
43. Add in the Parent Ring, Then Take the Quotient Class
Use the parent ring Z_20 with residues 0 through 19 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=17, b=15. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).
View answer and explanation
(1, 3, 12, 0, [a]+[b]=[a+b])
Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (1, 3, 12, 0, [a]+[b]=[a+b]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-add · candidate
44. Add in the Parent Ring, Then Take the Quotient Class
Use the parent ring Z_25 with residues 0 through 24 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=18, b=12. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).
View answer and explanation
(3, 2, 5, 0, [a]+[b]=[a+b])
Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (3, 2, 5, 0, [a]+[b]=[a+b]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-add · candidate
45. Add in the Parent Ring, Then Take the Quotient Class
Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=19, b=9. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).
View answer and explanation
(1, 3, 28, 4, [a]+[b]=[a+b])
Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (1, 3, 28, 4, [a]+[b]=[a+b]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-add · candidate
46. Add in the Parent Ring, Then Take the Quotient Class
Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=8, b=5. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).
View answer and explanation
(0, 1, 1, 1, [a]+[b]=[a+b])
Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (0, 1, 1, 1, [a]+[b]=[a+b]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-add · candidate
47. Add in the Parent Ring, Then Take the Quotient Class
Use the parent ring Z_18 with residues 0 through 17 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=3, b=7. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).
View answer and explanation
(0, 1, 10, 1, [a]+[b]=[a+b])
Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (0, 1, 10, 1, [a]+[b]=[a+b]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-add · candidate
48. Add in the Parent Ring, Then Take the Quotient Class
Use the parent ring Z_24 with residues 0 through 23 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=22, b=21. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).
View answer and explanation
(2, 1, 19, 3, [a]+[b]=[a+b])
Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (2, 1, 19, 3, [a]+[b]=[a+b]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-add · candidate
49. Add in the Parent Ring, Then Take the Quotient Class
Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=23, b=17. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).
View answer and explanation
(3, 2, 10, 0, [a]+[b]=[a+b])
Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (3, 2, 10, 0, [a]+[b]=[a+b]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-add · candidate
50. Add in the Parent Ring, Then Take the Quotient Class
Use the parent ring Z_36 with residues 0 through 35 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=24, b=13. Give (class_a,class_b,parent_result,result_class,[a]+[b]=[a+b]).
View answer and explanation
(0, 1, 1, 1, [a]+[b]=[a+b])
Compute a+b modulo n for the parent result, then reduce it modulo d for its class. These two output fields must not be conflated. Representative independence is a separate operation obligation. The exact result, retaining the requested order, is (0, 1, 1, 1, [a]+[b]=[a+b]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-add · candidate
51. Quotient Multiplication Must Respect the Same Ideal
Use the parent ring Z_4 with residues 0 through 3 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=0, b=1. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).
View answer and explanation
(0, 1, 0, 0, [a][b]=[ab])
Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (0, 1, 0, 0, [a][b]=[ab]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate
52. Quotient Multiplication Must Respect the Same Ideal
Use the parent ring Z_6 with residues 0 through 5 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=1, b=3. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).
View answer and explanation
(1, 0, 3, 0, [a][b]=[ab])
Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (1, 0, 3, 0, [a][b]=[ab]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate
53. Quotient Multiplication Must Respect the Same Ideal
Use the parent ring Z_8 with residues 0 through 7 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=2, b=5. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).
View answer and explanation
(2, 1, 2, 2, [a][b]=[ab])
Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (2, 1, 2, 2, [a][b]=[ab]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate
54. Quotient Multiplication Must Respect the Same Ideal
Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=3, b=7. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).
View answer and explanation
(3, 2, 1, 1, [a][b]=[ab])
Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (3, 2, 1, 1, [a][b]=[ab]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate
55. Quotient Multiplication Must Respect the Same Ideal
Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=4, b=9. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).
View answer and explanation
(4, 3, 0, 0, [a][b]=[ab])
Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (4, 3, 0, 0, [a][b]=[ab]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate
56. Quotient Multiplication Must Respect the Same Ideal
Use the parent ring Z_6 with residues 0 through 5 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=5, b=5. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).
View answer and explanation
(1, 1, 1, 1, [a][b]=[ab])
Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (1, 1, 1, 1, [a][b]=[ab]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate
57. Quotient Multiplication Must Respect the Same Ideal
Use the parent ring Z_9 with residues 0 through 8 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=6, b=4. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).
View answer and explanation
(0, 1, 6, 0, [a][b]=[ab])
Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (0, 1, 6, 0, [a][b]=[ab]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate
58. Quotient Multiplication Must Respect the Same Ideal
Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=7, b=3. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).
View answer and explanation
(3, 3, 9, 1, [a][b]=[ab])
Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (3, 3, 9, 1, [a][b]=[ab]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate
59. Quotient Multiplication Must Respect the Same Ideal
Use the parent ring Z_15 with residues 0 through 14 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=8, b=2. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).
View answer and explanation
(3, 2, 1, 1, [a][b]=[ab])
Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (3, 2, 1, 1, [a][b]=[ab]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate
60. Quotient Multiplication Must Respect the Same Ideal
Use the parent ring Z_18 with residues 0 through 17 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=9, b=1. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).
View answer and explanation
(3, 1, 9, 3, [a][b]=[ab])
Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (3, 1, 9, 3, [a][b]=[ab]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate
61. Quotient Multiplication Must Respect the Same Ideal
Use the parent ring Z_8 with residues 0 through 7 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=2, b=5. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).
View answer and explanation
(0, 1, 2, 0, [a][b]=[ab])
Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (0, 1, 2, 0, [a][b]=[ab]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate
62. Quotient Multiplication Must Respect the Same Ideal
Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=11, b=11. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).
View answer and explanation
(2, 2, 1, 1, [a][b]=[ab])
Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (2, 2, 1, 1, [a][b]=[ab]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate
63. Quotient Multiplication Must Respect the Same Ideal
Use the parent ring Z_16 with residues 0 through 15 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=12, b=9. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).
View answer and explanation
(0, 1, 12, 0, [a][b]=[ab])
Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (0, 1, 12, 0, [a][b]=[ab]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate
64. Quotient Multiplication Must Respect the Same Ideal
Use the parent ring Z_20 with residues 0 through 19 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=13, b=7. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).
View answer and explanation
(3, 2, 11, 1, [a][b]=[ab])
Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (3, 2, 11, 1, [a][b]=[ab]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate
65. Quotient Multiplication Must Respect the Same Ideal
Use the parent ring Z_24 with residues 0 through 23 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=14, b=5. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).
View answer and explanation
(2, 5, 22, 4, [a][b]=[ab])
Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (2, 5, 22, 4, [a][b]=[ab]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate
66. Quotient Multiplication Must Respect the Same Ideal
Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=5, b=1. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).
View answer and explanation
(1, 1, 5, 1, [a][b]=[ab])
Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (1, 1, 5, 1, [a][b]=[ab]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate
67. Quotient Multiplication Must Respect the Same Ideal
Use the parent ring Z_15 with residues 0 through 14 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=1, b=3. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).
View answer and explanation
(1, 0, 3, 0, [a][b]=[ab])
Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (1, 0, 3, 0, [a][b]=[ab]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate
68. Quotient Multiplication Must Respect the Same Ideal
Use the parent ring Z_20 with residues 0 through 19 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=17, b=15. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).
View answer and explanation
(1, 3, 15, 3, [a][b]=[ab])
Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (1, 3, 15, 3, [a][b]=[ab]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate
69. Quotient Multiplication Must Respect the Same Ideal
Use the parent ring Z_25 with residues 0 through 24 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=18, b=12. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).
View answer and explanation
(3, 2, 16, 1, [a][b]=[ab])
Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (3, 2, 16, 1, [a][b]=[ab]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate
70. Quotient Multiplication Must Respect the Same Ideal
Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=19, b=9. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).
View answer and explanation
(1, 3, 21, 3, [a][b]=[ab])
Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (1, 3, 21, 3, [a][b]=[ab]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate
71. Quotient Multiplication Must Respect the Same Ideal
Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=8, b=5. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).
View answer and explanation
(0, 1, 4, 0, [a][b]=[ab])
Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (0, 1, 4, 0, [a][b]=[ab]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate
72. Quotient Multiplication Must Respect the Same Ideal
Use the parent ring Z_18 with residues 0 through 17 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=3, b=7. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).
View answer and explanation
(0, 1, 3, 0, [a][b]=[ab])
Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (0, 1, 3, 0, [a][b]=[ab]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate
73. Quotient Multiplication Must Respect the Same Ideal
Use the parent ring Z_24 with residues 0 through 23 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=22, b=21. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).
View answer and explanation
(2, 1, 6, 2, [a][b]=[ab])
Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (2, 1, 6, 2, [a][b]=[ab]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate
74. Quotient Multiplication Must Respect the Same Ideal
Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=23, b=17. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).
View answer and explanation
(3, 2, 1, 1, [a][b]=[ab])
Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (3, 2, 1, 1, [a][b]=[ab]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate
75. Quotient Multiplication Must Respect the Same Ideal
Use the parent ring Z_36 with residues 0 through 35 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Parent representatives a=24, b=13. Give (class_a,class_b,parent_result,result_class,[a][b]=[ab]).
View answer and explanation
(0, 1, 24, 0, [a][b]=[ab])
Compute a*b modulo n and then its class modulo d. The two-sided ideal premise makes multiplication well-defined; arbitrary equivalence classes do not automatically inherit it. The exact result, retaining the requested order, is (0, 1, 24, 0, [a][b]=[ab]).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-multiply · candidate
76. Changing Both Representatives Must Preserve Both Operations
Use the parent ring Z_4 with residues 0 through 3 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=0, b=1, a_prime=2, b_prime=3 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).
View answer and explanation
([0, 1, 2, 3], [1, 1], [0, 0], Yes, Yes, Yes)
Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([0, 1, 2, 3], [1, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate
77. Changing Both Representatives Must Preserve Both Operations
Use the parent ring Z_6 with residues 0 through 5 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=1, b=0, a_prime=4, b_prime=3 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).
View answer and explanation
([1, 0, 4, 3], [1, 1], [0, 0], Yes, Yes, Yes)
Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([1, 0, 4, 3], [1, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate
78. Changing Both Representatives Must Preserve Both Operations
Use the parent ring Z_8 with residues 0 through 7 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=2, b=1, a_prime=6, b_prime=5 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).
View answer and explanation
([2, 1, 6, 5], [3, 3], [2, 2], Yes, Yes, Yes)
Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([2, 1, 6, 5], [3, 3], [2, 2], Yes, Yes, Yes).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate
79. Changing Both Representatives Must Preserve Both Operations
Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=3, b=2, a_prime=8, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).
View answer and explanation
([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes)
Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate
80. Changing Both Representatives Must Preserve Both Operations
Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=4, b=3, a_prime=10, b_prime=9 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).
View answer and explanation
([4, 3, 10, 9], [1, 1], [0, 0], Yes, Yes, Yes)
Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([4, 3, 10, 9], [1, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate
81. Changing Both Representatives Must Preserve Both Operations
Use the parent ring Z_6 with residues 0 through 5 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=1, b=1, a_prime=3, b_prime=3 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).
View answer and explanation
([1, 1, 3, 3], [0, 0], [1, 1], Yes, Yes, Yes)
Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([1, 1, 3, 3], [0, 0], [1, 1], Yes, Yes, Yes).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate
82. Changing Both Representatives Must Preserve Both Operations
Use the parent ring Z_9 with residues 0 through 8 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=0, b=1, a_prime=3, b_prime=4 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).
View answer and explanation
([0, 1, 3, 4], [1, 1], [0, 0], Yes, Yes, Yes)
Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([0, 1, 3, 4], [1, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate
83. Changing Both Representatives Must Preserve Both Operations
Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=3, b=3, a_prime=7, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).
View answer and explanation
([3, 3, 7, 7], [2, 2], [1, 1], Yes, Yes, Yes)
Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([3, 3, 7, 7], [2, 2], [1, 1], Yes, Yes, Yes).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate
84. Changing Both Representatives Must Preserve Both Operations
Use the parent ring Z_15 with residues 0 through 14 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=3, b=2, a_prime=8, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).
View answer and explanation
([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes)
Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate
85. Changing Both Representatives Must Preserve Both Operations
Use the parent ring Z_18 with residues 0 through 17 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=3, b=1, a_prime=9, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).
View answer and explanation
([3, 1, 9, 7], [4, 4], [3, 3], Yes, Yes, Yes)
Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([3, 1, 9, 7], [4, 4], [3, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate
86. Changing Both Representatives Must Preserve Both Operations
Use the parent ring Z_8 with residues 0 through 7 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=0, b=1, a_prime=2, b_prime=3 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).
View answer and explanation
([0, 1, 2, 3], [1, 1], [0, 0], Yes, Yes, Yes)
Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([0, 1, 2, 3], [1, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate
87. Changing Both Representatives Must Preserve Both Operations
Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=2, b=2, a_prime=5, b_prime=5 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).
View answer and explanation
([2, 2, 5, 5], [1, 1], [1, 1], Yes, Yes, Yes)
Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([2, 2, 5, 5], [1, 1], [1, 1], Yes, Yes, Yes).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate
88. Changing Both Representatives Must Preserve Both Operations
Use the parent ring Z_16 with residues 0 through 15 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=0, b=1, a_prime=4, b_prime=5 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).
View answer and explanation
([0, 1, 4, 5], [1, 1], [0, 0], Yes, Yes, Yes)
Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([0, 1, 4, 5], [1, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate
89. Changing Both Representatives Must Preserve Both Operations
Use the parent ring Z_20 with residues 0 through 19 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=3, b=2, a_prime=8, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).
View answer and explanation
([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes)
Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate
90. Changing Both Representatives Must Preserve Both Operations
Use the parent ring Z_24 with residues 0 through 23 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=2, b=5, a_prime=8, b_prime=11 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).
View answer and explanation
([2, 5, 8, 11], [1, 1], [4, 4], Yes, Yes, Yes)
Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([2, 5, 8, 11], [1, 1], [4, 4], Yes, Yes, Yes).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate
91. Changing Both Representatives Must Preserve Both Operations
Use the parent ring Z_10 with residues 0 through 9 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=1, b=1, a_prime=3, b_prime=3 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).
View answer and explanation
([1, 1, 3, 3], [0, 0], [1, 1], Yes, Yes, Yes)
Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([1, 1, 3, 3], [0, 0], [1, 1], Yes, Yes, Yes).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate
92. Changing Both Representatives Must Preserve Both Operations
Use the parent ring Z_15 with residues 0 through 14 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=1, b=0, a_prime=4, b_prime=3 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).
View answer and explanation
([1, 0, 4, 3], [1, 1], [0, 0], Yes, Yes, Yes)
Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([1, 0, 4, 3], [1, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate
93. Changing Both Representatives Must Preserve Both Operations
Use the parent ring Z_20 with residues 0 through 19 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=1, b=3, a_prime=5, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).
View answer and explanation
([1, 3, 5, 7], [0, 0], [3, 3], Yes, Yes, Yes)
Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([1, 3, 5, 7], [0, 0], [3, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate
94. Changing Both Representatives Must Preserve Both Operations
Use the parent ring Z_25 with residues 0 through 24 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=3, b=2, a_prime=8, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).
View answer and explanation
([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes)
Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate
95. Changing Both Representatives Must Preserve Both Operations
Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=1, b=3, a_prime=7, b_prime=9 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).
View answer and explanation
([1, 3, 7, 9], [4, 4], [3, 3], Yes, Yes, Yes)
Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([1, 3, 7, 9], [4, 4], [3, 3], Yes, Yes, Yes).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate
96. Changing Both Representatives Must Preserve Both Operations
Use the parent ring Z_12 with residues 0 through 11 and the proper two-sided ideal I of multiples of d=2, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=0, b=1, a_prime=2, b_prime=3 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).
View answer and explanation
([0, 1, 2, 3], [1, 1], [0, 0], Yes, Yes, Yes)
Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([0, 1, 2, 3], [1, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate
97. Changing Both Representatives Must Preserve Both Operations
Use the parent ring Z_18 with residues 0 through 17 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=0, b=1, a_prime=3, b_prime=4 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).
View answer and explanation
([0, 1, 3, 4], [1, 1], [0, 0], Yes, Yes, Yes)
Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([0, 1, 3, 4], [1, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate
98. Changing Both Representatives Must Preserve Both Operations
Use the parent ring Z_24 with residues 0 through 23 and the proper two-sided ideal I of multiples of d=4, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=2, b=1, a_prime=6, b_prime=5 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).
View answer and explanation
([2, 1, 6, 5], [3, 3], [2, 2], Yes, Yes, Yes)
Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([2, 1, 6, 5], [3, 3], [2, 2], Yes, Yes, Yes).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate
99. Changing Both Representatives Must Preserve Both Operations
Use the parent ring Z_30 with residues 0 through 29 and the proper two-sided ideal I of multiples of d=5, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=3, b=2, a_prime=8, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).
View answer and explanation
([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes)
Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([3, 2, 8, 7], [0, 0], [1, 1], Yes, Yes, Yes).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate
100. Changing Both Representatives Must Preserve Both Operations
Use the parent ring Z_36 with residues 0 through 35 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=0, b=1, a_prime=6, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).
View answer and explanation
([0, 1, 6, 7], [1, 1], [0, 0], Yes, Yes, Yes)
Compute both representative choices under addition and multiplication, retaining the two result-class arrays in choice order. A mismatch of input cosets is rejected rather than treated as a failed well-definedness theorem. The exact result, retaining the requested order, is ([0, 1, 6, 7], [1, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Quotient ring of a finite ring · qf-mod31-quotring-well-defined · candidate
Canonical question JSONL · English JSONL · Validation report