# 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.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0100/)

Chinese remainder theorem · qf-mod35-crt-noncoprime-boundary · candidate



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