# Question 6347: Positive Integer Hypotenuse

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0011-06347 · q-20261009-advance-20261013-r01-pythagorean-theorem-0047



## Original question and givens

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=66, b=88, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.


- G001 — Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=66, b=88, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-euclid-pythagorean-theorem](https://ocme.evemisslab.com/en/mko/mko-euclid-pythagorean-theorem/)





## Step-by-step derivation

S001 · impliesBefore

Euclidean right-side model;positive leg a=66

After

use the supplied right-angle premise:a²+b²=c²

Conditions: none

Use the declared Euclidean right-side model with positive legs.This finite numerical check is not a new universal proof of the Pythagorean theorem or an arbitrary affine-geometric bridge.

Step identifiers and dependenciesRule: pythagorean_right_side_premise · Uses: G001





S002 · calculationBefore

known leg a squared := square(66)

After

known leg a squared = 4356

Conditions: none

Square the complete known positive leg a.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

leg b squared := square(88)

After

leg b squared = 7744

Conditions: none

Square the other positive leg b separately.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

sum of leg squares := sum(4356, 7744)

After

sum of leg squares = 12100

Conditions: none

The sum of leg squares is the model's squared hypotenuse,not the hypotenuse itself.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

hypotenuse principal root squared := square(110)

After

hypotenuse principal root squared = 12100

Conditions: none

Square the integer root to audit the positive-perfect-square premise,without rounded floating roots.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · impliesBefore

root=110>0;root²=12100=required square

After

unique positive length=110

Conditions: none

Matching the square still requires a positive length;a negative root is not a side length,and0 violates this nondegenerate positive-leg protocol.

Step identifiers and dependenciesRule: positive_integer_length_root · Uses: G001, S005









## Reverse checks


- S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S005 · premise_and_result_audit — Recompute hypotenuse principal root squared with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute sum of leg squares with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute leg b squared with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute known leg a squared with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

110

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: b9bc71709e63d54b4ef665261c324afeea71d0e98a87e5c9c5b85117cc0edbd4
- candidate record SHA-256: 2153f996131798f5c9d24afa640e4a4f7ae0ea305bb55db9efd2ff894cb9e4bf
- semantic review receipt SHA-256: b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80
- question-first witness SHA-256: 93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0047.json)