Normalized CRT Solutions, Product Uniqueness and Noncoprime Compatibility
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batch-20261014-advance-20261104-r05-chinese-remainder-theorem-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.
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1. Solve a Pairwise-Coprime Pair after Normalizing Both Residues
Use the complete integer congruence system with positive distinct moduli=[3, 5] and aligned original residues=[0, 1] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).
View answer and explanation
([3, 5], [0, 1], 6, 15, Yes)
Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 5], [0, 1], 6, 15, Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-pair · candidate
2. Solve a Pairwise-Coprime Pair after Normalizing Both Residues
Use the complete integer congruence system with positive distinct moduli=[3, 7] and aligned original residues=[1, 3] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).
View answer and explanation
([3, 7], [1, 3], 10, 21, Yes)
Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 7], [1, 3], 10, 21, Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-pair · candidate
3. Solve a Pairwise-Coprime Pair after Normalizing Both Residues
Use the complete integer congruence system with positive distinct moduli=[3, 8] and aligned original residues=[2, 5] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).
View answer and explanation
([3, 8], [2, 5], 5, 24, Yes)
Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 8], [2, 5], 5, 24, Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-pair · candidate
4. Solve a Pairwise-Coprime Pair after Normalizing Both Residues
Use the complete integer congruence system with positive distinct moduli=[3, 10] and aligned original residues=[3, 7] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).
View answer and explanation
([3, 10], [0, 7], 27, 30, Yes)
Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 10], [0, 7], 27, 30, Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-pair · candidate
5. Solve a Pairwise-Coprime Pair after Normalizing Both Residues
Use the complete integer congruence system with positive distinct moduli=[3, 11] and aligned original residues=[4, 9] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).
View answer and explanation
([3, 11], [1, 9], 31, 33, Yes)
Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 11], [1, 9], 31, 33, Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-pair · candidate
6. Solve a Pairwise-Coprime Pair after Normalizing Both Residues
Use the complete integer congruence system with positive distinct moduli=[3, 13] and aligned original residues=[5, 11] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).
View answer and explanation
([3, 13], [2, 11], 11, 39, Yes)
Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 13], [2, 11], 11, 39, Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-pair · candidate
7. Solve a Pairwise-Coprime Pair after Normalizing Both Residues
Use the complete integer congruence system with positive distinct moduli=[3, 14] and aligned original residues=[6, 13] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).
View answer and explanation
([3, 14], [0, 13], 27, 42, Yes)
Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 14], [0, 13], 27, 42, Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-pair · candidate
8. Solve a Pairwise-Coprime Pair after Normalizing Both Residues
Use the complete integer congruence system with positive distinct moduli=[3, 16] and aligned original residues=[7, 15] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).
View answer and explanation
([3, 16], [1, 15], 31, 48, Yes)
Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 16], [1, 15], 31, 48, Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-pair · candidate
9. Solve a Pairwise-Coprime Pair after Normalizing Both Residues
Use the complete integer congruence system with positive distinct moduli=[3, 17] and aligned original residues=[8, 17] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).
View answer and explanation
([3, 17], [2, 0], 17, 51, Yes)
Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 17], [2, 0], 17, 51, Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-pair · candidate
10. Solve a Pairwise-Coprime Pair after Normalizing Both Residues
Use the complete integer congruence system with positive distinct moduli=[3, 19] and aligned original residues=[9, 19] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).
View answer and explanation
([3, 19], [0, 0], 0, 57, Yes)
Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 19], [0, 0], 0, 57, Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-pair · candidate
11. Solve a Pairwise-Coprime Pair after Normalizing Both Residues
Use the complete integer congruence system with positive distinct moduli=[3, 20] and aligned original residues=[10, 21] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).
View answer and explanation
([3, 20], [1, 1], 1, 60, Yes)
Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 20], [1, 1], 1, 60, Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-pair · candidate
12. Solve a Pairwise-Coprime Pair after Normalizing Both Residues
Use the complete integer congruence system with positive distinct moduli=[3, 22] and aligned original residues=[11, 23] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).
View answer and explanation
([3, 22], [2, 1], 23, 66, Yes)
Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 22], [2, 1], 23, 66, Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-pair · candidate
13. Solve a Pairwise-Coprime Pair after Normalizing Both Residues
Use the complete integer congruence system with positive distinct moduli=[3, 23] and aligned original residues=[12, 25] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).
View answer and explanation
([3, 23], [0, 2], 48, 69, Yes)
Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 23], [0, 2], 48, 69, Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-pair · candidate
14. Solve a Pairwise-Coprime Pair after Normalizing Both Residues
Use the complete integer congruence system with positive distinct moduli=[3, 25] and aligned original residues=[13, 27] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).
View answer and explanation
([3, 25], [1, 2], 52, 75, Yes)
Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 25], [1, 2], 52, 75, Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-pair · candidate
15. Solve a Pairwise-Coprime Pair after Normalizing Both Residues
Use the complete integer congruence system with positive distinct moduli=[3, 26] and aligned original residues=[14, 29] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).
View answer and explanation
([3, 26], [2, 3], 29, 78, Yes)
Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 26], [2, 3], 29, 78, Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-pair · candidate
16. Solve a Pairwise-Coprime Pair after Normalizing Both Residues
Use the complete integer congruence system with positive distinct moduli=[3, 28] and aligned original residues=[15, 31] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).
View answer and explanation
([3, 28], [0, 3], 3, 84, Yes)
Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 28], [0, 3], 3, 84, Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-pair · candidate
17. Solve a Pairwise-Coprime Pair after Normalizing Both Residues
Use the complete integer congruence system with positive distinct moduli=[3, 29] and aligned original residues=[16, 33] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).
View answer and explanation
([3, 29], [1, 4], 4, 87, Yes)
Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 29], [1, 4], 4, 87, Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-pair · candidate
18. Solve a Pairwise-Coprime Pair after Normalizing Both Residues
Use the complete integer congruence system with positive distinct moduli=[3, 31] and aligned original residues=[17, 35] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).
View answer and explanation
([3, 31], [2, 4], 35, 93, Yes)
Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 31], [2, 4], 35, 93, Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-pair · candidate
19. Solve a Pairwise-Coprime Pair after Normalizing Both Residues
Use the complete integer congruence system with positive distinct moduli=[3, 32] and aligned original residues=[18, 37] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).
View answer and explanation
([3, 32], [0, 5], 69, 96, Yes)
Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 32], [0, 5], 69, 96, Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-pair · candidate
20. Solve a Pairwise-Coprime Pair after Normalizing Both Residues
Use the complete integer congruence system with positive distinct moduli=[3, 34] and aligned original residues=[19, 39] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).
View answer and explanation
([3, 34], [1, 5], 73, 102, Yes)
Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 34], [1, 5], 73, 102, Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-pair · candidate
21. Solve a Pairwise-Coprime Pair after Normalizing Both Residues
Use the complete integer congruence system with positive distinct moduli=[3, 35] and aligned original residues=[20, 41] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).
View answer and explanation
([3, 35], [2, 6], 41, 105, Yes)
Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 35], [2, 6], 41, 105, Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-pair · candidate
22. Solve a Pairwise-Coprime Pair after Normalizing Both Residues
Use the complete integer congruence system with positive distinct moduli=[3, 37] and aligned original residues=[21, 43] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).
View answer and explanation
([3, 37], [0, 6], 6, 111, Yes)
Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 37], [0, 6], 6, 111, Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-pair · candidate
23. Solve a Pairwise-Coprime Pair after Normalizing Both Residues
Use the complete integer congruence system with positive distinct moduli=[3, 38] and aligned original residues=[22, 45] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).
View answer and explanation
([3, 38], [1, 7], 7, 114, Yes)
Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 38], [1, 7], 7, 114, Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-pair · candidate
24. Solve a Pairwise-Coprime Pair after Normalizing Both Residues
Use the complete integer congruence system with positive distinct moduli=[3, 40] and aligned original residues=[23, 47] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).
View answer and explanation
([3, 40], [2, 7], 47, 120, Yes)
Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 40], [2, 7], 47, 120, Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-pair · candidate
25. Solve a Pairwise-Coprime Pair after Normalizing Both Residues
Use the complete integer congruence system with positive distinct moduli=[3, 41] and aligned original residues=[24, 49] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Exactly two moduli are required here. Give (moduli, normalized_residues, solution, product, solved).
View answer and explanation
([3, 41], [0, 8], 90, 123, Yes)
Find the least nonnegative simultaneous solution, give the product modulus, and check both congruences. Signed or out-of-range residue inputs are normalized rather than changing the congruence classes. The exact result, retaining the requested order, is ([3, 41], [0, 8], 90, 123, Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-pair · candidate
26. Every Congruence Must Pass in the Same CRT System
Use the complete integer congruence system with positive distinct moduli=[3, 5, 7] and aligned original residues=[0, 1, 2] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).
View answer and explanation
([3, 5, 7], [0, 1, 2], 51, 105, [Yes, Yes, Yes], Yes)
Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 7], [0, 1, 2], 51, 105, [Yes, Yes, Yes], Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-system · candidate
27. Every Congruence Must Pass in the Same CRT System
Use the complete integer congruence system with positive distinct moduli=[3, 5, 8] and aligned original residues=[1, 3, 5] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).
View answer and explanation
([3, 5, 8], [1, 3, 5], 13, 120, [Yes, Yes, Yes], Yes)
Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 8], [1, 3, 5], 13, 120, [Yes, Yes, Yes], Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-system · candidate
28. Every Congruence Must Pass in the Same CRT System
Use the complete integer congruence system with positive distinct moduli=[3, 5, 11] and aligned original residues=[2, 5, 8] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).
View answer and explanation
([3, 5, 11], [2, 0, 8], 140, 165, [Yes, Yes, Yes], Yes)
Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 11], [2, 0, 8], 140, 165, [Yes, Yes, Yes], Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-system · candidate
29. Every Congruence Must Pass in the Same CRT System
Use the complete integer congruence system with positive distinct moduli=[3, 5, 13] and aligned original residues=[3, 7, 11] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).
View answer and explanation
([3, 5, 13], [0, 2, 11], 102, 195, [Yes, Yes, Yes], Yes)
Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 13], [0, 2, 11], 102, 195, [Yes, Yes, Yes], Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-system · candidate
30. Every Congruence Must Pass in the Same CRT System
Use the complete integer congruence system with positive distinct moduli=[3, 5, 14] and aligned original residues=[4, 9, 14] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).
View answer and explanation
([3, 5, 14], [1, 4, 0], 154, 210, [Yes, Yes, Yes], Yes)
Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 14], [1, 4, 0], 154, 210, [Yes, Yes, Yes], Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-system · candidate
31. Every Congruence Must Pass in the Same CRT System
Use the complete integer congruence system with positive distinct moduli=[3, 5, 16] and aligned original residues=[5, 11, 17] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).
View answer and explanation
([3, 5, 16], [2, 1, 1], 161, 240, [Yes, Yes, Yes], Yes)
Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 16], [2, 1, 1], 161, 240, [Yes, Yes, Yes], Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-system · candidate
32. Every Congruence Must Pass in the Same CRT System
Use the complete integer congruence system with positive distinct moduli=[3, 5, 17] and aligned original residues=[6, 13, 20] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).
View answer and explanation
([3, 5, 17], [0, 3, 3], 3, 255, [Yes, Yes, Yes], Yes)
Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 17], [0, 3, 3], 3, 255, [Yes, Yes, Yes], Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-system · candidate
33. Every Congruence Must Pass in the Same CRT System
Use the complete integer congruence system with positive distinct moduli=[3, 5, 19] and aligned original residues=[7, 15, 23] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).
View answer and explanation
([3, 5, 19], [1, 0, 4], 175, 285, [Yes, Yes, Yes], Yes)
Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 19], [1, 0, 4], 175, 285, [Yes, Yes, Yes], Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-system · candidate
34. Every Congruence Must Pass in the Same CRT System
Use the complete integer congruence system with positive distinct moduli=[3, 5, 22] and aligned original residues=[8, 17, 26] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).
View answer and explanation
([3, 5, 22], [2, 2, 4], 92, 330, [Yes, Yes, Yes], Yes)
Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 22], [2, 2, 4], 92, 330, [Yes, Yes, Yes], Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-system · candidate
35. Every Congruence Must Pass in the Same CRT System
Use the complete integer congruence system with positive distinct moduli=[3, 5, 23] and aligned original residues=[9, 19, 29] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).
View answer and explanation
([3, 5, 23], [0, 4, 6], 144, 345, [Yes, Yes, Yes], Yes)
Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 23], [0, 4, 6], 144, 345, [Yes, Yes, Yes], Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-system · candidate
36. Every Congruence Must Pass in the Same CRT System
Use the complete integer congruence system with positive distinct moduli=[3, 5, 26] and aligned original residues=[10, 21, 32] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).
View answer and explanation
([3, 5, 26], [1, 1, 6], 136, 390, [Yes, Yes, Yes], Yes)
Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 26], [1, 1, 6], 136, 390, [Yes, Yes, Yes], Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-system · candidate
37. Every Congruence Must Pass in the Same CRT System
Use the complete integer congruence system with positive distinct moduli=[3, 5, 28] and aligned original residues=[11, 23, 35] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).
View answer and explanation
([3, 5, 28], [2, 3, 7], 203, 420, [Yes, Yes, Yes], Yes)
Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 28], [2, 3, 7], 203, 420, [Yes, Yes, Yes], Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-system · candidate
38. Every Congruence Must Pass in the Same CRT System
Use the complete integer congruence system with positive distinct moduli=[3, 5, 29] and aligned original residues=[12, 25, 38] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).
View answer and explanation
([3, 5, 29], [0, 0, 9], 270, 435, [Yes, Yes, Yes], Yes)
Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 29], [0, 0, 9], 270, 435, [Yes, Yes, Yes], Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-system · candidate
39. Every Congruence Must Pass in the Same CRT System
Use the complete integer congruence system with positive distinct moduli=[3, 5, 31] and aligned original residues=[13, 27, 41] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).
View answer and explanation
([3, 5, 31], [1, 2, 10], 382, 465, [Yes, Yes, Yes], Yes)
Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 31], [1, 2, 10], 382, 465, [Yes, Yes, Yes], Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-system · candidate
40. Every Congruence Must Pass in the Same CRT System
Use the complete integer congruence system with positive distinct moduli=[3, 5, 32] and aligned original residues=[14, 29, 44] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).
View answer and explanation
([3, 5, 32], [2, 4, 12], 44, 480, [Yes, Yes, Yes], Yes)
Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 32], [2, 4, 12], 44, 480, [Yes, Yes, Yes], Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-system · candidate
41. Every Congruence Must Pass in the Same CRT System
Use the complete integer congruence system with positive distinct moduli=[3, 5, 34] and aligned original residues=[15, 31, 47] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).
View answer and explanation
([3, 5, 34], [0, 1, 13], 81, 510, [Yes, Yes, Yes], Yes)
Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 34], [0, 1, 13], 81, 510, [Yes, Yes, Yes], Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-system · candidate
42. Every Congruence Must Pass in the Same CRT System
Use the complete integer congruence system with positive distinct moduli=[3, 5, 37] and aligned original residues=[16, 33, 50] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).
View answer and explanation
([3, 5, 37], [1, 3, 13], 13, 555, [Yes, Yes, Yes], Yes)
Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 37], [1, 3, 13], 13, 555, [Yes, Yes, Yes], Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-system · candidate
43. Every Congruence Must Pass in the Same CRT System
Use the complete integer congruence system with positive distinct moduli=[3, 5, 38] and aligned original residues=[17, 35, 53] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).
View answer and explanation
([3, 5, 38], [2, 0, 15], 395, 570, [Yes, Yes, Yes], Yes)
Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 38], [2, 0, 15], 395, 570, [Yes, Yes, Yes], Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-system · candidate
44. Every Congruence Must Pass in the Same CRT System
Use the complete integer congruence system with positive distinct moduli=[3, 5, 41] and aligned original residues=[18, 37, 56] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).
View answer and explanation
([3, 5, 41], [0, 2, 15], 507, 615, [Yes, Yes, Yes], Yes)
Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 41], [0, 2, 15], 507, 615, [Yes, Yes, Yes], Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-system · candidate
45. Every Congruence Must Pass in the Same CRT System
Use the complete integer congruence system with positive distinct moduli=[3, 5, 43] and aligned original residues=[19, 39, 59] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).
View answer and explanation
([3, 5, 43], [1, 4, 16], 274, 645, [Yes, Yes, Yes], Yes)
Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 43], [1, 4, 16], 274, 645, [Yes, Yes, Yes], Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-system · candidate
46. Every Congruence Must Pass in the Same CRT System
Use the complete integer congruence system with positive distinct moduli=[3, 5, 44] and aligned original residues=[20, 41, 62] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).
View answer and explanation
([3, 5, 44], [2, 1, 18], 326, 660, [Yes, Yes, Yes], Yes)
Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 44], [2, 1, 18], 326, 660, [Yes, Yes, Yes], Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-system · candidate
47. Every Congruence Must Pass in the Same CRT System
Use the complete integer congruence system with positive distinct moduli=[3, 5, 46] and aligned original residues=[21, 43, 65] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).
View answer and explanation
([3, 5, 46], [0, 3, 19], 663, 690, [Yes, Yes, Yes], Yes)
Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 46], [0, 3, 19], 663, 690, [Yes, Yes, Yes], Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-system · candidate
48. Every Congruence Must Pass in the Same CRT System
Use the complete integer congruence system with positive distinct moduli=[3, 5, 47] and aligned original residues=[22, 45, 68] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).
View answer and explanation
([3, 5, 47], [1, 0, 21], 115, 705, [Yes, Yes, Yes], Yes)
Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 47], [1, 0, 21], 115, 705, [Yes, Yes, Yes], Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-system · candidate
49. Every Congruence Must Pass in the Same CRT System
Use the complete integer congruence system with positive distinct moduli=[3, 5, 49] and aligned original residues=[23, 47, 71] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).
View answer and explanation
([3, 5, 49], [2, 2, 22], 512, 735, [Yes, Yes, Yes], Yes)
Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 49], [2, 2, 22], 512, 735, [Yes, Yes, Yes], Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-system · candidate
50. Every Congruence Must Pass in the Same CRT System
Use the complete integer congruence system with positive distinct moduli=[3, 5, 52] and aligned original residues=[24, 49, 74] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).
View answer and explanation
([3, 5, 52], [0, 4, 22], 594, 780, [Yes, Yes, Yes], Yes)
Return the least nonnegative solution and the full product modulus, then retain each congruence check in original input order. The all-equations certificate is not established by one passing congruence. The exact result, retaining the requested order, is ([3, 5, 52], [0, 4, 22], 594, 780, [Yes, Yes, Yes], Yes).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-solve-system · candidate
51. Uniqueness Is a Residue-Class Claim, Not a Unique Integer
Use the complete integer congruence system with positive distinct moduli=[3, 5] and aligned original residues=[0, 1] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=21. Give (solution, other, product, same_residue_class, boundary).
View answer and explanation
(6, 21, 15, Yes, unique_modulo_product_not_unique_integer)
Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (6, 21, 15, Yes, unique_modulo_product_not_unique_integer).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-uniqueness-mod-product · candidate
52. Uniqueness Is a Residue-Class Claim, Not a Unique Integer
Use the complete integer congruence system with positive distinct moduli=[3, 7] and aligned original residues=[1, 3] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=31. Give (solution, other, product, same_residue_class, boundary).
View answer and explanation
(10, 31, 21, Yes, unique_modulo_product_not_unique_integer)
Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (10, 31, 21, Yes, unique_modulo_product_not_unique_integer).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-uniqueness-mod-product · candidate
53. Uniqueness Is a Residue-Class Claim, Not a Unique Integer
Use the complete integer congruence system with positive distinct moduli=[3, 8] and aligned original residues=[2, 5] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=29. Give (solution, other, product, same_residue_class, boundary).
View answer and explanation
(5, 29, 24, Yes, unique_modulo_product_not_unique_integer)
Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (5, 29, 24, Yes, unique_modulo_product_not_unique_integer).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-uniqueness-mod-product · candidate
54. Uniqueness Is a Residue-Class Claim, Not a Unique Integer
Use the complete integer congruence system with positive distinct moduli=[3, 10] and aligned original residues=[3, 7] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=57. Give (solution, other, product, same_residue_class, boundary).
View answer and explanation
(27, 57, 30, Yes, unique_modulo_product_not_unique_integer)
Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (27, 57, 30, Yes, unique_modulo_product_not_unique_integer).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-uniqueness-mod-product · candidate
55. Uniqueness Is a Residue-Class Claim, Not a Unique Integer
Use the complete integer congruence system with positive distinct moduli=[3, 11] and aligned original residues=[4, 9] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=64. Give (solution, other, product, same_residue_class, boundary).
View answer and explanation
(31, 64, 33, Yes, unique_modulo_product_not_unique_integer)
Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (31, 64, 33, Yes, unique_modulo_product_not_unique_integer).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-uniqueness-mod-product · candidate
56. Uniqueness Is a Residue-Class Claim, Not a Unique Integer
Use the complete integer congruence system with positive distinct moduli=[3, 13] and aligned original residues=[5, 11] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=50. Give (solution, other, product, same_residue_class, boundary).
View answer and explanation
(11, 50, 39, Yes, unique_modulo_product_not_unique_integer)
Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (11, 50, 39, Yes, unique_modulo_product_not_unique_integer).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-uniqueness-mod-product · candidate
57. Uniqueness Is a Residue-Class Claim, Not a Unique Integer
Use the complete integer congruence system with positive distinct moduli=[3, 14] and aligned original residues=[6, 13] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=69. Give (solution, other, product, same_residue_class, boundary).
View answer and explanation
(27, 69, 42, Yes, unique_modulo_product_not_unique_integer)
Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (27, 69, 42, Yes, unique_modulo_product_not_unique_integer).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-uniqueness-mod-product · candidate
58. Uniqueness Is a Residue-Class Claim, Not a Unique Integer
Use the complete integer congruence system with positive distinct moduli=[3, 16] and aligned original residues=[7, 15] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=79. Give (solution, other, product, same_residue_class, boundary).
View answer and explanation
(31, 79, 48, Yes, unique_modulo_product_not_unique_integer)
Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (31, 79, 48, Yes, unique_modulo_product_not_unique_integer).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-uniqueness-mod-product · candidate
59. Uniqueness Is a Residue-Class Claim, Not a Unique Integer
Use the complete integer congruence system with positive distinct moduli=[3, 17] and aligned original residues=[8, 17] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=68. Give (solution, other, product, same_residue_class, boundary).
View answer and explanation
(17, 68, 51, Yes, unique_modulo_product_not_unique_integer)
Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (17, 68, 51, Yes, unique_modulo_product_not_unique_integer).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-uniqueness-mod-product · candidate
60. Uniqueness Is a Residue-Class Claim, Not a Unique Integer
Use the complete integer congruence system with positive distinct moduli=[3, 19] and aligned original residues=[9, 19] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=57. Give (solution, other, product, same_residue_class, boundary).
View answer and explanation
(0, 57, 57, Yes, unique_modulo_product_not_unique_integer)
Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (0, 57, 57, Yes, unique_modulo_product_not_unique_integer).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-uniqueness-mod-product · candidate
61. Uniqueness Is a Residue-Class Claim, Not a Unique Integer
Use the complete integer congruence system with positive distinct moduli=[3, 20] and aligned original residues=[10, 21] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=61. Give (solution, other, product, same_residue_class, boundary).
View answer and explanation
(1, 61, 60, Yes, unique_modulo_product_not_unique_integer)
Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (1, 61, 60, Yes, unique_modulo_product_not_unique_integer).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-uniqueness-mod-product · candidate
62. Uniqueness Is a Residue-Class Claim, Not a Unique Integer
Use the complete integer congruence system with positive distinct moduli=[3, 22] and aligned original residues=[11, 23] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=89. Give (solution, other, product, same_residue_class, boundary).
View answer and explanation
(23, 89, 66, Yes, unique_modulo_product_not_unique_integer)
Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (23, 89, 66, Yes, unique_modulo_product_not_unique_integer).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-uniqueness-mod-product · candidate
63. Uniqueness Is a Residue-Class Claim, Not a Unique Integer
Use the complete integer congruence system with positive distinct moduli=[3, 23] and aligned original residues=[12, 25] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=117. Give (solution, other, product, same_residue_class, boundary).
View answer and explanation
(48, 117, 69, Yes, unique_modulo_product_not_unique_integer)
Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (48, 117, 69, Yes, unique_modulo_product_not_unique_integer).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-uniqueness-mod-product · candidate
64. Uniqueness Is a Residue-Class Claim, Not a Unique Integer
Use the complete integer congruence system with positive distinct moduli=[3, 25] and aligned original residues=[13, 27] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=127. Give (solution, other, product, same_residue_class, boundary).
View answer and explanation
(52, 127, 75, Yes, unique_modulo_product_not_unique_integer)
Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (52, 127, 75, Yes, unique_modulo_product_not_unique_integer).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-uniqueness-mod-product · candidate
65. Uniqueness Is a Residue-Class Claim, Not a Unique Integer
Use the complete integer congruence system with positive distinct moduli=[3, 26] and aligned original residues=[14, 29] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=107. Give (solution, other, product, same_residue_class, boundary).
View answer and explanation
(29, 107, 78, Yes, unique_modulo_product_not_unique_integer)
Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (29, 107, 78, Yes, unique_modulo_product_not_unique_integer).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-uniqueness-mod-product · candidate
66. Uniqueness Is a Residue-Class Claim, Not a Unique Integer
Use the complete integer congruence system with positive distinct moduli=[3, 28] and aligned original residues=[15, 31] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=87. Give (solution, other, product, same_residue_class, boundary).
View answer and explanation
(3, 87, 84, Yes, unique_modulo_product_not_unique_integer)
Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (3, 87, 84, Yes, unique_modulo_product_not_unique_integer).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-uniqueness-mod-product · candidate
67. Uniqueness Is a Residue-Class Claim, Not a Unique Integer
Use the complete integer congruence system with positive distinct moduli=[3, 29] and aligned original residues=[16, 33] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=91. Give (solution, other, product, same_residue_class, boundary).
View answer and explanation
(4, 91, 87, Yes, unique_modulo_product_not_unique_integer)
Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (4, 91, 87, Yes, unique_modulo_product_not_unique_integer).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-uniqueness-mod-product · candidate
68. Uniqueness Is a Residue-Class Claim, Not a Unique Integer
Use the complete integer congruence system with positive distinct moduli=[3, 31] and aligned original residues=[17, 35] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=128. Give (solution, other, product, same_residue_class, boundary).
View answer and explanation
(35, 128, 93, Yes, unique_modulo_product_not_unique_integer)
Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (35, 128, 93, Yes, unique_modulo_product_not_unique_integer).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-uniqueness-mod-product · candidate
69. Uniqueness Is a Residue-Class Claim, Not a Unique Integer
Use the complete integer congruence system with positive distinct moduli=[3, 32] and aligned original residues=[18, 37] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=165. Give (solution, other, product, same_residue_class, boundary).
View answer and explanation
(69, 165, 96, Yes, unique_modulo_product_not_unique_integer)
Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (69, 165, 96, Yes, unique_modulo_product_not_unique_integer).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-uniqueness-mod-product · candidate
70. Uniqueness Is a Residue-Class Claim, Not a Unique Integer
Use the complete integer congruence system with positive distinct moduli=[3, 34] and aligned original residues=[19, 39] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=175. Give (solution, other, product, same_residue_class, boundary).
View answer and explanation
(73, 175, 102, Yes, unique_modulo_product_not_unique_integer)
Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (73, 175, 102, Yes, unique_modulo_product_not_unique_integer).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-uniqueness-mod-product · candidate
71. Uniqueness Is a Residue-Class Claim, Not a Unique Integer
Use the complete integer congruence system with positive distinct moduli=[3, 35] and aligned original residues=[20, 41] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=146. Give (solution, other, product, same_residue_class, boundary).
View answer and explanation
(41, 146, 105, Yes, unique_modulo_product_not_unique_integer)
Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (41, 146, 105, Yes, unique_modulo_product_not_unique_integer).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-uniqueness-mod-product · candidate
72. Uniqueness Is a Residue-Class Claim, Not a Unique Integer
Use the complete integer congruence system with positive distinct moduli=[3, 37] and aligned original residues=[21, 43] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=117. Give (solution, other, product, same_residue_class, boundary).
View answer and explanation
(6, 117, 111, Yes, unique_modulo_product_not_unique_integer)
Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (6, 117, 111, Yes, unique_modulo_product_not_unique_integer).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-uniqueness-mod-product · candidate
73. Uniqueness Is a Residue-Class Claim, Not a Unique Integer
Use the complete integer congruence system with positive distinct moduli=[3, 38] and aligned original residues=[22, 45] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=121. Give (solution, other, product, same_residue_class, boundary).
View answer and explanation
(7, 121, 114, Yes, unique_modulo_product_not_unique_integer)
Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (7, 121, 114, Yes, unique_modulo_product_not_unique_integer).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-uniqueness-mod-product · candidate
74. Uniqueness Is a Residue-Class Claim, Not a Unique Integer
Use the complete integer congruence system with positive distinct moduli=[3, 40] and aligned original residues=[23, 47] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=167. Give (solution, other, product, same_residue_class, boundary).
View answer and explanation
(47, 167, 120, Yes, unique_modulo_product_not_unique_integer)
Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (47, 167, 120, Yes, unique_modulo_product_not_unique_integer).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-uniqueness-mod-product · candidate
75. Uniqueness Is a Residue-Class Claim, Not a Unique Integer
Use the complete integer congruence system with positive distinct moduli=[3, 41] and aligned original residues=[24, 49] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Pairwise-coprime system; compare arbitrary integer other=213. Give (solution, other, product, same_residue_class, boundary).
View answer and explanation
(90, 213, 123, Yes, unique_modulo_product_not_unique_integer)
Test whether other differs from the least solution by a multiple of the product. The comparison integer is not presumed to solve the system. A No therefore does not refute CRT uniqueness; distinct integers in the solution class differ by product multiples. The exact result, retaining the requested order, is (90, 213, 123, Yes, unique_modulo_product_not_unique_integer).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-uniqueness-mod-product · candidate
76. A Noncoprime System Can Be Compatible or Incompatible
Use the complete integer congruence system with positive distinct moduli=[4, 6] and aligned original residues=[1, 3] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).
View answer and explanation
([4, 6], [1, 3], Yes, 9, 12, noncoprime_not_automatically_inconsistent)
Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([4, 6], [1, 3], Yes, 9, 12, noncoprime_not_automatically_inconsistent).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate
77. A Noncoprime System Can Be Compatible or Incompatible
Use the complete integer congruence system with positive distinct moduli=[6, 9] and aligned original residues=[1, 4] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).
View answer and explanation
([6, 9], [1, 4], Yes, 13, 18, noncoprime_not_automatically_inconsistent)
Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([6, 9], [1, 4], Yes, 13, 18, noncoprime_not_automatically_inconsistent).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate
78. A Noncoprime System Can Be Compatible or Incompatible
Use the complete integer congruence system with positive distinct moduli=[8, 12] and aligned original residues=[1, 5] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).
View answer and explanation
([8, 12], [1, 5], Yes, 17, 24, noncoprime_not_automatically_inconsistent)
Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([8, 12], [1, 5], Yes, 17, 24, noncoprime_not_automatically_inconsistent).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate
79. A Noncoprime System Can Be Compatible or Incompatible
Use the complete integer congruence system with positive distinct moduli=[10, 15] and aligned original residues=[1, 6] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).
View answer and explanation
([10, 15], [1, 6], Yes, 21, 30, noncoprime_not_automatically_inconsistent)
Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([10, 15], [1, 6], Yes, 21, 30, noncoprime_not_automatically_inconsistent).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate
80. A Noncoprime System Can Be Compatible or Incompatible
Use the complete integer congruence system with positive distinct moduli=[12, 18] and aligned original residues=[1, 7] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).
View answer and explanation
([12, 18], [1, 7], Yes, 25, 36, noncoprime_not_automatically_inconsistent)
Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([12, 18], [1, 7], Yes, 25, 36, noncoprime_not_automatically_inconsistent).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate
81. A Noncoprime System Can Be Compatible or Incompatible
Use the complete integer congruence system with positive distinct moduli=[14, 21] and aligned original residues=[1, 8] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).
View answer and explanation
([14, 21], [1, 8], Yes, 29, 42, noncoprime_not_automatically_inconsistent)
Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([14, 21], [1, 8], Yes, 29, 42, noncoprime_not_automatically_inconsistent).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate
82. A Noncoprime System Can Be Compatible or Incompatible
Use the complete integer congruence system with positive distinct moduli=[16, 24] and aligned original residues=[1, 9] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).
View answer and explanation
([16, 24], [1, 9], Yes, 33, 48, noncoprime_not_automatically_inconsistent)
Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([16, 24], [1, 9], Yes, 33, 48, noncoprime_not_automatically_inconsistent).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate
83. A Noncoprime System Can Be Compatible or Incompatible
Use the complete integer congruence system with positive distinct moduli=[18, 27] and aligned original residues=[1, 10] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).
View answer and explanation
([18, 27], [1, 10], Yes, 37, 54, noncoprime_not_automatically_inconsistent)
Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([18, 27], [1, 10], Yes, 37, 54, noncoprime_not_automatically_inconsistent).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate
84. A Noncoprime System Can Be Compatible or Incompatible
Use the complete integer congruence system with positive distinct moduli=[20, 30] and aligned original residues=[1, 11] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).
View answer and explanation
([20, 30], [1, 11], Yes, 41, 60, noncoprime_not_automatically_inconsistent)
Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([20, 30], [1, 11], Yes, 41, 60, noncoprime_not_automatically_inconsistent).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate
85. A Noncoprime System Can Be Compatible or Incompatible
Use the complete integer congruence system with positive distinct moduli=[22, 33] and aligned original residues=[1, 12] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).
View answer and explanation
([22, 33], [1, 12], Yes, 45, 66, noncoprime_not_automatically_inconsistent)
Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([22, 33], [1, 12], Yes, 45, 66, noncoprime_not_automatically_inconsistent).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate
86. A Noncoprime System Can Be Compatible or Incompatible
Use the complete integer congruence system with positive distinct moduli=[24, 36] and aligned original residues=[1, 13] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).
View answer and explanation
([24, 36], [1, 13], Yes, 49, 72, noncoprime_not_automatically_inconsistent)
Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([24, 36], [1, 13], Yes, 49, 72, noncoprime_not_automatically_inconsistent).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate
87. A Noncoprime System Can Be Compatible or Incompatible
Use the complete integer congruence system with positive distinct moduli=[26, 39] and aligned original residues=[1, 14] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).
View answer and explanation
([26, 39], [1, 14], Yes, 53, 78, noncoprime_not_automatically_inconsistent)
Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([26, 39], [1, 14], Yes, 53, 78, noncoprime_not_automatically_inconsistent).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate
88. A Noncoprime System Can Be Compatible or Incompatible
Use the complete integer congruence system with positive distinct moduli=[28, 42] and aligned original residues=[1, 15] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).
View answer and explanation
([28, 42], [1, 15], Yes, 57, 84, noncoprime_not_automatically_inconsistent)
Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([28, 42], [1, 15], Yes, 57, 84, noncoprime_not_automatically_inconsistent).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate
89. A Noncoprime System Can Be Compatible or Incompatible
Use the complete integer congruence system with positive distinct moduli=[30, 45] and aligned original residues=[1, 16] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).
View answer and explanation
([30, 45], [1, 16], Yes, 61, 90, noncoprime_not_automatically_inconsistent)
Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([30, 45], [1, 16], Yes, 61, 90, noncoprime_not_automatically_inconsistent).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate
90. A Noncoprime System Can Be Compatible or Incompatible
Use the complete integer congruence system with positive distinct moduli=[32, 48] and aligned original residues=[1, 17] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).
View answer and explanation
([32, 48], [1, 17], Yes, 65, 96, noncoprime_not_automatically_inconsistent)
Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([32, 48], [1, 17], Yes, 65, 96, noncoprime_not_automatically_inconsistent).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate
91. A Noncoprime System Can Be Compatible or Incompatible
Use the complete integer congruence system with positive distinct moduli=[34, 51] and aligned original residues=[1, 18] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).
View answer and explanation
([34, 51], [1, 18], Yes, 69, 102, noncoprime_not_automatically_inconsistent)
Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([34, 51], [1, 18], Yes, 69, 102, noncoprime_not_automatically_inconsistent).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate
92. A Noncoprime System Can Be Compatible or Incompatible
Use the complete integer congruence system with positive distinct moduli=[36, 54] and aligned original residues=[1, 19] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).
View answer and explanation
([36, 54], [1, 19], Yes, 73, 108, noncoprime_not_automatically_inconsistent)
Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([36, 54], [1, 19], Yes, 73, 108, noncoprime_not_automatically_inconsistent).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate
93. A Noncoprime System Can Be Compatible or Incompatible
Use the complete integer congruence system with positive distinct moduli=[38, 57] and aligned original residues=[1, 20] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).
View answer and explanation
([38, 57], [1, 20], Yes, 77, 114, noncoprime_not_automatically_inconsistent)
Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([38, 57], [1, 20], Yes, 77, 114, noncoprime_not_automatically_inconsistent).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate
94. A Noncoprime System Can Be Compatible or Incompatible
Use the complete integer congruence system with positive distinct moduli=[40, 60] and aligned original residues=[1, 21] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).
View answer and explanation
([40, 60], [1, 21], Yes, 81, 120, noncoprime_not_automatically_inconsistent)
Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([40, 60], [1, 21], Yes, 81, 120, noncoprime_not_automatically_inconsistent).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate
95. A Noncoprime System Can Be Compatible or Incompatible
Use the complete integer congruence system with positive distinct moduli=[42, 63] and aligned original residues=[1, 22] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).
View answer and explanation
([42, 63], [1, 22], Yes, 85, 126, noncoprime_not_automatically_inconsistent)
Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([42, 63], [1, 22], Yes, 85, 126, noncoprime_not_automatically_inconsistent).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate
96. A Noncoprime System Can Be Compatible or Incompatible
Use the complete integer congruence system with positive distinct moduli=[44, 66] and aligned original residues=[1, 23] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).
View answer and explanation
([44, 66], [1, 23], Yes, 89, 132, noncoprime_not_automatically_inconsistent)
Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([44, 66], [1, 23], Yes, 89, 132, noncoprime_not_automatically_inconsistent).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate
97. A Noncoprime System Can Be Compatible or Incompatible
Use the complete integer congruence system with positive distinct moduli=[46, 69] and aligned original residues=[1, 24] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).
View answer and explanation
([46, 69], [1, 24], Yes, 93, 138, noncoprime_not_automatically_inconsistent)
Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([46, 69], [1, 24], Yes, 93, 138, noncoprime_not_automatically_inconsistent).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate
98. A Noncoprime System Can Be Compatible or Incompatible
Use the complete integer congruence system with positive distinct moduli=[48, 72] and aligned original residues=[1, 25] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).
View answer and explanation
([48, 72], [1, 25], Yes, 97, 144, noncoprime_not_automatically_inconsistent)
Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([48, 72], [1, 25], Yes, 97, 144, noncoprime_not_automatically_inconsistent).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate
99. A Noncoprime System Can Be Compatible or Incompatible
Use the complete integer congruence system with positive distinct moduli=[50, 75] and aligned original residues=[1, 26] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).
View answer and explanation
([50, 75], [1, 26], Yes, 101, 150, noncoprime_not_automatically_inconsistent)
Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([50, 75], [1, 26], Yes, 101, 150, noncoprime_not_automatically_inconsistent).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate
100. A Noncoprime System Can Be Compatible or Incompatible
Use the complete integer congruence system with positive distinct moduli=[52, 78] and aligned original residues=[1, 27] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. At least one modulus pair must be noncoprime. Give (moduli, normalized_residues, compatible, solution_or_undefined, lcm_period, boundary).
View answer and explanation
([52, 78], [1, 27], Yes, 105, 156, noncoprime_not_automatically_inconsistent)
Test every residue difference modulo the corresponding gcd. When compatible return the least solution and lcm period; when incompatible retain the literal undefined solution field. The product is not generally the solution period in this boundary case. The exact result, retaining the requested order, is ([52, 78], [1, 27], Yes, 105, 156, noncoprime_not_automatically_inconsistent).
Reviewed step-by-step derivation
Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate
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