# Prime-Power Extraction, Product Additivity and Sum Cancellation

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations · 2026-10-15 · 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. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=2, nonzero integer a=2. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation(2, 2, 2, 1, 1, [2], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, 2, 2, 1, 1, [2], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0001/)

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate





## 2. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=3, nonzero integer a=-18. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation(3, -18, 18, 2, 2, [3, 9], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (3, -18, 18, 2, 2, [3, 9], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0002/)

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate





## 3. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=5, nonzero integer a=375. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation(5, 375, 375, 3, 3, [5, 25, 125], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (5, 375, 375, 3, 3, [5, 25, 125], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0003/)

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate





## 4. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=2, nonzero integer a=-8. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation(2, -8, 8, 3, 1, [2, 4, 8], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, -8, 8, 3, 1, [2, 4, 8], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0004/)

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate





## 5. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=3, nonzero integer a=45. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation(3, 45, 45, 2, 5, [3, 9], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (3, 45, 45, 2, 5, [3, 9], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0005/)

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate





## 6. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=5, nonzero integer a=-750. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation(5, -750, 750, 3, 6, [5, 25, 125], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (5, -750, 750, 3, 6, [5, 25, 125], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0006/)

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate





## 7. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=2, nonzero integer a=14. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation(2, 14, 14, 1, 7, [2], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, 14, 14, 1, 7, [2], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0007/)

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate





## 8. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=3, nonzero integer a=-9. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation(3, -9, 9, 2, 1, [3, 9], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (3, -9, 9, 2, 1, [3, 9], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0008/)

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate





## 9. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=5, nonzero integer a=250. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation(5, 250, 250, 3, 2, [5, 25, 125], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (5, 250, 250, 3, 2, [5, 25, 125], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0009/)

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate





## 10. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=2, nonzero integer a=-6. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation(2, -6, 6, 1, 3, [2], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, -6, 6, 1, 3, [2], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0010/)

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate





## 11. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=3, nonzero integer a=36. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation(3, 36, 36, 2, 4, [3, 9], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (3, 36, 36, 2, 4, [3, 9], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0011/)

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate





## 12. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=5, nonzero integer a=-625. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation(5, -625, 625, 4, 1, [5, 25, 125, 625], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (5, -625, 625, 4, 1, [5, 25, 125, 625], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0012/)

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate





## 13. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=2, nonzero integer a=12. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation(2, 12, 12, 2, 3, [2, 4], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, 12, 12, 2, 3, [2, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0013/)

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate





## 14. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=3, nonzero integer a=-63. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation(3, -63, 63, 2, 7, [3, 9], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (3, -63, 63, 2, 7, [3, 9], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0014/)

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate





## 15. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=5, nonzero integer a=125. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation(5, 125, 125, 3, 1, [5, 25, 125], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (5, 125, 125, 3, 1, [5, 25, 125], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0015/)

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate





## 16. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=2, nonzero integer a=-4. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation(2, -4, 4, 2, 1, [2, 4], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, -4, 4, 2, 1, [2, 4], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0016/)

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate





## 17. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=3, nonzero integer a=27. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation(3, 27, 27, 3, 1, [3, 9, 27], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (3, 27, 27, 3, 1, [3, 9, 27], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0017/)

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate





## 18. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=5, nonzero integer a=-500. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation(5, -500, 500, 3, 4, [5, 25, 125], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (5, -500, 500, 3, 4, [5, 25, 125], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0018/)

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate





## 19. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=2, nonzero integer a=10. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation(2, 10, 10, 1, 5, [2], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, 10, 10, 1, 5, [2], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0019/)

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate





## 20. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=3, nonzero integer a=-54. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation(3, -54, 54, 3, 2, [3, 9, 27], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (3, -54, 54, 3, 2, [3, 9, 27], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0020/)

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate





## 21. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=5, nonzero integer a=875. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation(5, 875, 875, 3, 7, [5, 25, 125], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (5, 875, 875, 3, 7, [5, 25, 125], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0021/)

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate





## 22. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=2, nonzero integer a=-2. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation(2, -2, 2, 1, 1, [2], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, -2, 2, 1, 1, [2], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0022/)

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate





## 23. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=3, nonzero integer a=18. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation(3, 18, 18, 2, 2, [3, 9], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (3, 18, 18, 2, 2, [3, 9], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0023/)

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate





## 24. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=5, nonzero integer a=-375. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation(5, -375, 375, 3, 3, [5, 25, 125], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (5, -375, 375, 3, 3, [5, 25, 125], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0024/)

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate





## 25. The Highest Dividing Prime Power Is Extracted from the Absolute Integer

Use prime p=2, nonzero integer a=8. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).

View answer and explanation(2, 8, 8, 3, 1, [2, 4, 8], Yes, Yes)

Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, 8, 8, 3, 1, [2, 4, 8], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0025/)

p-adic valuation of integers · qf-mod49-pval-factor-extraction · candidate





## 26. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=2, integers a=1, b=-4, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation(2, 1, -4, 0, 2, -4, 2, 2, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (2, 1, -4, 0, 2, -4, 2, 2, Yes, infinity_plus_k_equals_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0026/)

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate





## 27. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=3, integers a=6, b=-27, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation(3, 6, -27, 1, 3, -162, 4, 4, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (3, 6, -27, 1, 3, -162, 4, 4, Yes, infinity_plus_k_equals_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0027/)

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate





## 28. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=5, integers a=75, b=-4, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation(5, 75, -4, 2, 0, -300, 2, 2, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (5, 75, -4, 2, 0, -300, 2, 2, Yes, infinity_plus_k_equals_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0028/)

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate





## 29. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=7, integers a=4, b=-35, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation(7, 4, -35, 0, 1, -140, 1, 1, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (7, 4, -35, 0, 1, -140, 1, 1, Yes, infinity_plus_k_equals_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0029/)

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate





## 30. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=11, integers a=55, b=-726, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation(11, 55, -726, 1, 2, -39930, 3, 3, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (11, 55, -726, 1, 2, -39930, 3, 3, Yes, infinity_plus_k_equals_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0030/)

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate





## 31. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=13, integers a=1014, b=-7, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation(13, 1014, -7, 2, 0, -7098, 2, 2, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (13, 1014, -7, 2, 0, -7098, 2, 2, Yes, infinity_plus_k_equals_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0031/)

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate





## 32. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=2, integers a=7, b=-16, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation(2, 7, -16, 0, 4, -112, 4, 4, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (2, 7, -16, 0, 4, -112, 4, 4, Yes, infinity_plus_k_equals_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0032/)

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate





## 33. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=3, integers a=24, b=-81, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation(3, 24, -81, 1, 4, -1944, 5, 5, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (3, 24, -81, 1, 4, -1944, 5, 5, Yes, infinity_plus_k_equals_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0033/)

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate





## 34. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=5, integers a=225, b=-10, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation(5, 225, -10, 2, 1, -2250, 3, 3, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (5, 225, -10, 2, 1, -2250, 3, 3, Yes, infinity_plus_k_equals_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0034/)

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate





## 35. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=7, integers a=10, b=-77, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation(7, 10, -77, 0, 1, -770, 1, 1, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (7, 10, -77, 0, 1, -770, 1, 1, Yes, infinity_plus_k_equals_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0035/)

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate





## 36. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=11, integers a=121, b=-1452, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation(11, 121, -1452, 2, 2, -175692, 4, 4, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (11, 121, -1452, 2, 2, -175692, 4, 4, Yes, infinity_plus_k_equals_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0036/)

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate





## 37. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=13, integers a=2028, b=-13, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation(13, 2028, -13, 2, 1, -26364, 3, 3, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (13, 2028, -13, 2, 1, -26364, 3, 3, Yes, infinity_plus_k_equals_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0037/)

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate





## 38. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=2, integers a=13, b=-28, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation(2, 13, -28, 0, 2, -364, 2, 2, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (2, 13, -28, 0, 2, -364, 2, 2, Yes, infinity_plus_k_equals_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0038/)

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate





## 39. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=3, integers a=42, b=-135, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation(3, 42, -135, 1, 3, -5670, 4, 4, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (3, 42, -135, 1, 3, -5670, 4, 4, Yes, infinity_plus_k_equals_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0039/)

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate





## 40. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=5, integers a=375, b=-16, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation(5, 375, -16, 3, 0, -6000, 3, 3, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (5, 375, -16, 3, 0, -6000, 3, 3, Yes, infinity_plus_k_equals_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0040/)

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate





## 41. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=7, integers a=16, b=-119, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation(7, 16, -119, 0, 1, -1904, 1, 1, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (7, 16, -119, 0, 1, -1904, 1, 1, Yes, infinity_plus_k_equals_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0041/)

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate





## 42. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=11, integers a=187, b=-2178, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation(11, 187, -2178, 1, 2, -407286, 3, 3, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (11, 187, -2178, 1, 2, -407286, 3, 3, Yes, infinity_plus_k_equals_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0042/)

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate





## 43. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=13, integers a=3042, b=-19, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation(13, 3042, -19, 2, 0, -57798, 2, 2, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (13, 3042, -19, 2, 0, -57798, 2, 2, Yes, infinity_plus_k_equals_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0043/)

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate





## 44. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=2, integers a=19, b=-40, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation(2, 19, -40, 0, 3, -760, 3, 3, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (2, 19, -40, 0, 3, -760, 3, 3, Yes, infinity_plus_k_equals_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0044/)

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate





## 45. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=3, integers a=60, b=-189, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation(3, 60, -189, 1, 3, -11340, 4, 4, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (3, 60, -189, 1, 3, -11340, 4, 4, Yes, infinity_plus_k_equals_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0045/)

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate





## 46. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=5, integers a=525, b=-22, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation(5, 525, -22, 2, 0, -11550, 2, 2, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (5, 525, -22, 2, 0, -11550, 2, 2, Yes, infinity_plus_k_equals_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0046/)

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate





## 47. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=7, integers a=22, b=-161, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation(7, 22, -161, 0, 1, -3542, 1, 1, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (7, 22, -161, 0, 1, -3542, 1, 1, Yes, infinity_plus_k_equals_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0047/)

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate





## 48. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=11, integers a=253, b=-2904, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation(11, 253, -2904, 1, 2, -734712, 3, 3, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (11, 253, -2904, 1, 2, -734712, 3, 3, Yes, infinity_plus_k_equals_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0048/)

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate





## 49. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=13, integers a=4056, b=-25, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation(13, 4056, -25, 2, 0, -101400, 2, 2, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (13, 4056, -25, 2, 0, -101400, 2, 2, Yes, infinity_plus_k_equals_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0049/)

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate





## 50. Integer Valuations Add under Products, Including the Fixed Zero Convention

Use prime p=2, integers a=25, b=-52, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).

View answer and explanation(2, 25, -52, 0, 2, -1300, 2, 2, Yes, infinity_plus_k_equals_infinity)

Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (2, 25, -52, 0, 2, -1300, 2, 2, Yes, infinity_plus_k_equals_infinity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0050/)

p-adic valuation of integers · qf-mod49-pval-product-additivity · candidate





## 51. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=2, nonzero summands a=2, b=4; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation(2, 2, 4, 1, 2, 6, 1, 1, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (2, 2, 4, 1, 2, 6, 1, 1, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0051/)

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate





## 52. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=3, nonzero summands a=18, b=27; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation(3, 18, 27, 2, 3, 45, 2, 2, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (3, 18, 27, 2, 3, 45, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0052/)

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate





## 53. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=5, nonzero summands a=375, b=4; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation(5, 375, 4, 3, 0, 379, 0, 0, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (5, 375, 4, 3, 0, 379, 0, 0, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0053/)

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate





## 54. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=7, nonzero summands a=28, b=35; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation(7, 28, 35, 1, 1, 63, 1, 1, Yes, No, not_guaranteed)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (7, 28, 35, 1, 1, 63, 1, 1, Yes, No, not_guaranteed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0054/)

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate





## 55. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=11, nonzero summands a=605, b=726; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation(11, 605, 726, 2, 2, 1331, 3, 2, Yes, No, not_guaranteed)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (11, 605, 726, 2, 2, 1331, 3, 2, Yes, No, not_guaranteed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0055/)

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate





## 56. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=13, nonzero summands a=13182, b=7; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation(13, 13182, 7, 3, 0, 13189, 0, 0, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (13, 13182, 7, 3, 0, 13189, 0, 0, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0056/)

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate





## 57. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=2, nonzero summands a=14, b=16; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation(2, 14, 16, 1, 4, 30, 1, 1, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (2, 14, 16, 1, 4, 30, 1, 1, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0057/)

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate





## 58. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=3, nonzero summands a=72, b=81; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation(3, 72, 81, 2, 4, 153, 2, 2, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (3, 72, 81, 2, 4, 153, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0058/)

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate





## 59. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=5, nonzero summands a=1125, b=10; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation(5, 1125, 10, 3, 1, 1135, 1, 1, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (5, 1125, 10, 3, 1, 1135, 1, 1, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0059/)

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate





## 60. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=7, nonzero summands a=70, b=77; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation(7, 70, 77, 1, 1, 147, 2, 1, Yes, No, not_guaranteed)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (7, 70, 77, 1, 1, 147, 2, 1, Yes, No, not_guaranteed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0060/)

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate





## 61. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=11, nonzero summands a=1331, b=1452; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation(11, 1331, 1452, 3, 2, 2783, 2, 2, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (11, 1331, 1452, 3, 2, 2783, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0061/)

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate





## 62. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=13, nonzero summands a=26364, b=13; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation(13, 26364, 13, 3, 1, 26377, 1, 1, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (13, 26364, 13, 3, 1, 26377, 1, 1, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0062/)

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate





## 63. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=2, nonzero summands a=26, b=28; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation(2, 26, 28, 1, 2, 54, 1, 1, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (2, 26, 28, 1, 2, 54, 1, 1, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0063/)

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate





## 64. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=3, nonzero summands a=126, b=135; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation(3, 126, 135, 2, 3, 261, 2, 2, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (3, 126, 135, 2, 3, 261, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0064/)

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate





## 65. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=5, nonzero summands a=1875, b=16; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation(5, 1875, 16, 4, 0, 1891, 0, 0, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (5, 1875, 16, 4, 0, 1891, 0, 0, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0065/)

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate





## 66. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=7, nonzero summands a=112, b=119; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation(7, 112, 119, 1, 1, 231, 1, 1, Yes, No, not_guaranteed)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (7, 112, 119, 1, 1, 231, 1, 1, Yes, No, not_guaranteed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0066/)

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate





## 67. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=11, nonzero summands a=2057, b=2178; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation(11, 2057, 2178, 2, 2, 4235, 2, 2, Yes, No, not_guaranteed)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (11, 2057, 2178, 2, 2, 4235, 2, 2, Yes, No, not_guaranteed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0067/)

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate





## 68. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=13, nonzero summands a=39546, b=19; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation(13, 39546, 19, 3, 0, 39565, 0, 0, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (13, 39546, 19, 3, 0, 39565, 0, 0, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0068/)

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate





## 69. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=2, nonzero summands a=38, b=40; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation(2, 38, 40, 1, 3, 78, 1, 1, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (2, 38, 40, 1, 3, 78, 1, 1, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0069/)

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate





## 70. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=3, nonzero summands a=180, b=189; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation(3, 180, 189, 2, 3, 369, 2, 2, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (3, 180, 189, 2, 3, 369, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0070/)

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate





## 71. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=5, nonzero summands a=2625, b=22; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation(5, 2625, 22, 3, 0, 2647, 0, 0, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (5, 2625, 22, 3, 0, 2647, 0, 0, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0071/)

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate





## 72. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=7, nonzero summands a=154, b=161; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation(7, 154, 161, 1, 1, 315, 1, 1, Yes, No, not_guaranteed)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (7, 154, 161, 1, 1, 315, 1, 1, Yes, No, not_guaranteed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0072/)

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate





## 73. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=11, nonzero summands a=2783, b=2904; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation(11, 2783, 2904, 2, 2, 5687, 2, 2, Yes, No, not_guaranteed)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (11, 2783, 2904, 2, 2, 5687, 2, 2, Yes, No, not_guaranteed).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0073/)

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate





## 74. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=13, nonzero summands a=52728, b=25; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation(13, 52728, 25, 3, 0, 52753, 0, 0, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (13, 52728, 25, 3, 0, 52753, 0, 0, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0074/)

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate





## 75. Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations

Use prime p=2, nonzero summands a=50, b=52; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).

valuation_sum may be infinity. When summand valuations agree, the last field is not_guaranteed; it does not assert that equality fails in every such instance.

View answer and explanation(2, 50, 52, 1, 2, 102, 1, 1, Yes, Yes, Yes)

Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (2, 50, 52, 1, 2, 102, 1, 1, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0075/)

p-adic valuation of integers · qf-mod49-pval-sum-ultrametric · candidate





## 76. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=zero_convention, base p=2, integer a=0. Give (mode, p, integer_a, valuation, convention).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation(zero_convention, 2, 0, infinity, single_serializable_zero_convention)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (zero_convention, 2, 0, infinity, single_serializable_zero_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0076/)

p-adic valuation of integers · qf-mod49-pval-boundary · candidate





## 77. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=nonprime_base, base p=6, integer a=2. Give (mode, base, integer_a, p_adic_valuation_qualification, boundary).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation(nonprime_base, 6, 2, No, not_called_p_adic_valuation)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (nonprime_base, 6, 2, No, not_called_p_adic_valuation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0077/)

p-adic valuation of integers · qf-mod49-pval-boundary · candidate





## 78. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=negative_normalization, base p=2, integer a=-12. Give (mode, p, integer_a, absolute_a, valuation, negative_integer_valid, scope).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation(negative_normalization, 2, -12, 12, 2, Yes, negative_integer_is_valid)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (negative_normalization, 2, -12, 12, 2, Yes, negative_integer_is_valid).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0078/)

p-adic valuation of integers · qf-mod49-pval-boundary · candidate





## 79. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=equal_valuations_cancellation, base p=2, integer a=8. Give (mode, p, a, opposite_a, valuation_a, valuation_opposite_a, valuation_sum, cancellation_scope, universal_sum_equality_claim).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation(equal_valuations_cancellation, 2, 8, -8, 3, 3, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (equal_valuations_cancellation, 2, 8, -8, 3, 3, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0079/)

p-adic valuation of integers · qf-mod49-pval-boundary · candidate





## 80. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=zero_convention, base p=3, integer a=0. Give (mode, p, integer_a, valuation, convention).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation(zero_convention, 3, 0, infinity, single_serializable_zero_convention)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (zero_convention, 3, 0, infinity, single_serializable_zero_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0080/)

p-adic valuation of integers · qf-mod49-pval-boundary · candidate





## 81. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=nonprime_base, base p=12, integer a=6. Give (mode, base, integer_a, p_adic_valuation_qualification, boundary).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation(nonprime_base, 12, 6, No, not_called_p_adic_valuation)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (nonprime_base, 12, 6, No, not_called_p_adic_valuation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0081/)

p-adic valuation of integers · qf-mod49-pval-boundary · candidate





## 82. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=negative_normalization, base p=3, integer a=-7. Give (mode, p, integer_a, absolute_a, valuation, negative_integer_valid, scope).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation(negative_normalization, 3, -7, 7, 0, Yes, negative_integer_is_valid)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (negative_normalization, 3, -7, 7, 0, Yes, negative_integer_is_valid).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0082/)

p-adic valuation of integers · qf-mod49-pval-boundary · candidate





## 83. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=equal_valuations_cancellation, base p=3, integer a=24. Give (mode, p, a, opposite_a, valuation_a, valuation_opposite_a, valuation_sum, cancellation_scope, universal_sum_equality_claim).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation(equal_valuations_cancellation, 3, 24, -24, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (equal_valuations_cancellation, 3, 24, -24, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0083/)

p-adic valuation of integers · qf-mod49-pval-boundary · candidate





## 84. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=zero_convention, base p=5, integer a=0. Give (mode, p, integer_a, valuation, convention).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation(zero_convention, 5, 0, infinity, single_serializable_zero_convention)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (zero_convention, 5, 0, infinity, single_serializable_zero_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0084/)

p-adic valuation of integers · qf-mod49-pval-boundary · candidate





## 85. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=nonprime_base, base p=18, integer a=10. Give (mode, base, integer_a, p_adic_valuation_qualification, boundary).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation(nonprime_base, 18, 10, No, not_called_p_adic_valuation)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (nonprime_base, 18, 10, No, not_called_p_adic_valuation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0085/)

p-adic valuation of integers · qf-mod49-pval-boundary · candidate





## 86. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=negative_normalization, base p=5, integer a=-55. Give (mode, p, integer_a, absolute_a, valuation, negative_integer_valid, scope).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation(negative_normalization, 5, -55, 55, 1, Yes, negative_integer_is_valid)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (negative_normalization, 5, -55, 55, 1, Yes, negative_integer_is_valid).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0086/)

p-adic valuation of integers · qf-mod49-pval-boundary · candidate





## 87. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=equal_valuations_cancellation, base p=5, integer a=60. Give (mode, p, a, opposite_a, valuation_a, valuation_opposite_a, valuation_sum, cancellation_scope, universal_sum_equality_claim).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation(equal_valuations_cancellation, 5, 60, -60, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (equal_valuations_cancellation, 5, 60, -60, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0087/)

p-adic valuation of integers · qf-mod49-pval-boundary · candidate





## 88. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=zero_convention, base p=7, integer a=0. Give (mode, p, integer_a, valuation, convention).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation(zero_convention, 7, 0, infinity, single_serializable_zero_convention)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (zero_convention, 7, 0, infinity, single_serializable_zero_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0088/)

p-adic valuation of integers · qf-mod49-pval-boundary · candidate





## 89. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=nonprime_base, base p=24, integer a=14. Give (mode, base, integer_a, p_adic_valuation_qualification, boundary).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation(nonprime_base, 24, 14, No, not_called_p_adic_valuation)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (nonprime_base, 24, 14, No, not_called_p_adic_valuation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0089/)

p-adic valuation of integers · qf-mod49-pval-boundary · candidate





## 90. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=negative_normalization, base p=7, integer a=-735. Give (mode, p, integer_a, absolute_a, valuation, negative_integer_valid, scope).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation(negative_normalization, 7, -735, 735, 2, Yes, negative_integer_is_valid)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (negative_normalization, 7, -735, 735, 2, Yes, negative_integer_is_valid).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0090/)

p-adic valuation of integers · qf-mod49-pval-boundary · candidate





## 91. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=equal_valuations_cancellation, base p=7, integer a=112. Give (mode, p, a, opposite_a, valuation_a, valuation_opposite_a, valuation_sum, cancellation_scope, universal_sum_equality_claim).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation(equal_valuations_cancellation, 7, 112, -112, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (equal_valuations_cancellation, 7, 112, -112, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0091/)

p-adic valuation of integers · qf-mod49-pval-boundary · candidate





## 92. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=zero_convention, base p=11, integer a=0. Give (mode, p, integer_a, valuation, convention).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation(zero_convention, 11, 0, infinity, single_serializable_zero_convention)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (zero_convention, 11, 0, infinity, single_serializable_zero_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0092/)

p-adic valuation of integers · qf-mod49-pval-boundary · candidate





## 93. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=nonprime_base, base p=28, integer a=18. Give (mode, base, integer_a, p_adic_valuation_qualification, boundary).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation(nonprime_base, 28, 18, No, not_called_p_adic_valuation)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (nonprime_base, 28, 18, No, not_called_p_adic_valuation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0093/)

p-adic valuation of integers · qf-mod49-pval-boundary · candidate





## 94. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=negative_normalization, base p=11, integer a=-19. Give (mode, p, integer_a, absolute_a, valuation, negative_integer_valid, scope).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation(negative_normalization, 11, -19, 19, 0, Yes, negative_integer_is_valid)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (negative_normalization, 11, -19, 19, 0, Yes, negative_integer_is_valid).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0094/)

p-adic valuation of integers · qf-mod49-pval-boundary · candidate





## 95. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=equal_valuations_cancellation, base p=11, integer a=220. Give (mode, p, a, opposite_a, valuation_a, valuation_opposite_a, valuation_sum, cancellation_scope, universal_sum_equality_claim).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation(equal_valuations_cancellation, 11, 220, -220, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (equal_valuations_cancellation, 11, 220, -220, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0095/)

p-adic valuation of integers · qf-mod49-pval-boundary · candidate





## 96. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=zero_convention, base p=13, integer a=0. Give (mode, p, integer_a, valuation, convention).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation(zero_convention, 13, 0, infinity, single_serializable_zero_convention)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (zero_convention, 13, 0, infinity, single_serializable_zero_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0096/)

p-adic valuation of integers · qf-mod49-pval-boundary · candidate





## 97. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=nonprime_base, base p=34, integer a=22. Give (mode, base, integer_a, p_adic_valuation_qualification, boundary).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation(nonprime_base, 34, 22, No, not_called_p_adic_valuation)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (nonprime_base, 34, 22, No, not_called_p_adic_valuation).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0097/)

p-adic valuation of integers · qf-mod49-pval-boundary · candidate





## 98. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=negative_normalization, base p=13, integer a=-299. Give (mode, p, integer_a, absolute_a, valuation, negative_integer_valid, scope).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation(negative_normalization, 13, -299, 299, 1, Yes, negative_integer_is_valid)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (negative_normalization, 13, -299, 299, 1, Yes, negative_integer_is_valid).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0098/)

p-adic valuation of integers · qf-mod49-pval-boundary · candidate





## 99. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=equal_valuations_cancellation, base p=13, integer a=312. Give (mode, p, a, opposite_a, valuation_a, valuation_opposite_a, valuation_sum, cancellation_scope, universal_sum_equality_claim).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation(equal_valuations_cancellation, 13, 312, -312, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (equal_valuations_cancellation, 13, 312, -312, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0099/)

p-adic valuation of integers · qf-mod49-pval-boundary · candidate





## 100. Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

Use mode=zero_convention, base p=17, integer a=0. Give (mode, p, integer_a, valuation, convention).

The four modes have5/5/7/9 fields. Negative integers are valid and normalized by absolute value; zero uses the single infinity sentinel.

View answer and explanation(zero_convention, 17, 0, infinity, single_serializable_zero_convention)

Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (zero_convention, 17, 0, infinity, single_serializable_zero_convention).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0100/)

p-adic valuation of integers · qf-mod49-pval-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations/validation-report.json)