# Positive Right-Triangle Sides and Square Residuals

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-20261009-advance-20261013-r01-pythagorean-theorem-foundations · 2026-10-09 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Ordered Leg Squares and Hypotenuse Square

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=1, b=1. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.

View answer and explanation(1, 1, 2)

Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (1, 1, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0001/)

Pythagorean theorem · qf-density13-pythagorean-leg-square-sum · candidate





## 2. Ordered Leg Squares and Hypotenuse Square

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=3, b=4. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.

View answer and explanation(9, 16, 25)

Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (9, 16, 25).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0002/)

Pythagorean theorem · qf-density13-pythagorean-leg-square-sum · candidate





## 3. Ordered Leg Squares and Hypotenuse Square

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=4, b=4. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.

View answer and explanation(16, 16, 32)

Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (16, 16, 32).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0003/)

Pythagorean theorem · qf-density13-pythagorean-leg-square-sum · candidate





## 4. Ordered Leg Squares and Hypotenuse Square

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=5, b=5. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.

View answer and explanation(25, 25, 50)

Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (25, 25, 50).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0004/)

Pythagorean theorem · qf-density13-pythagorean-leg-square-sum · candidate





## 5. Ordered Leg Squares and Hypotenuse Square

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=6, b=6. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.

View answer and explanation(36, 36, 72)

Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (36, 36, 72).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0005/)

Pythagorean theorem · qf-density13-pythagorean-leg-square-sum · candidate





## 6. Ordered Leg Squares and Hypotenuse Square

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=7, b=7. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.

View answer and explanation(49, 49, 98)

Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (49, 49, 98).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0006/)

Pythagorean theorem · qf-density13-pythagorean-leg-square-sum · candidate





## 7. Ordered Leg Squares and Hypotenuse Square

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=8, b=8. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.

View answer and explanation(64, 64, 128)

Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (64, 64, 128).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0007/)

Pythagorean theorem · qf-density13-pythagorean-leg-square-sum · candidate





## 8. Ordered Leg Squares and Hypotenuse Square

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=9, b=2. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.

View answer and explanation(81, 4, 85)

Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (81, 4, 85).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0008/)

Pythagorean theorem · qf-density13-pythagorean-leg-square-sum · candidate





## 9. Ordered Leg Squares and Hypotenuse Square

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=10, b=3. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.

View answer and explanation(100, 9, 109)

Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (100, 9, 109).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0009/)

Pythagorean theorem · qf-density13-pythagorean-leg-square-sum · candidate





## 10. Ordered Leg Squares and Hypotenuse Square

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=11, b=4. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.

View answer and explanation(121, 16, 137)

Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (121, 16, 137).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0010/)

Pythagorean theorem · qf-density13-pythagorean-leg-square-sum · candidate





## 11. Ordered Leg Squares and Hypotenuse Square

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=12, b=5. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.

View answer and explanation(144, 25, 169)

Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (144, 25, 169).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0011/)

Pythagorean theorem · qf-density13-pythagorean-leg-square-sum · candidate





## 12. Ordered Leg Squares and Hypotenuse Square

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=13, b=6. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.

View answer and explanation(169, 36, 205)

Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (169, 36, 205).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0012/)

Pythagorean theorem · qf-density13-pythagorean-leg-square-sum · candidate





## 13. Ordered Leg Squares and Hypotenuse Square

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=14, b=7. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.

View answer and explanation(196, 49, 245)

Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (196, 49, 245).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0013/)

Pythagorean theorem · qf-density13-pythagorean-leg-square-sum · candidate





## 14. Ordered Leg Squares and Hypotenuse Square

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=15, b=8. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.

View answer and explanation(225, 64, 289)

Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (225, 64, 289).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0014/)

Pythagorean theorem · qf-density13-pythagorean-leg-square-sum · candidate





## 15. Ordered Leg Squares and Hypotenuse Square

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=16, b=2. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.

View answer and explanation(256, 4, 260)

Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (256, 4, 260).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0015/)

Pythagorean theorem · qf-density13-pythagorean-leg-square-sum · candidate





## 16. Ordered Leg Squares and Hypotenuse Square

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=17, b=3. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.

View answer and explanation(289, 9, 298)

Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (289, 9, 298).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0016/)

Pythagorean theorem · qf-density13-pythagorean-leg-square-sum · candidate





## 17. Ordered Leg Squares and Hypotenuse Square

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=18, b=4. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.

View answer and explanation(324, 16, 340)

Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (324, 16, 340).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0017/)

Pythagorean theorem · qf-density13-pythagorean-leg-square-sum · candidate





## 18. Ordered Leg Squares and Hypotenuse Square

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=19, b=5. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.

View answer and explanation(361, 25, 386)

Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (361, 25, 386).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0018/)

Pythagorean theorem · qf-density13-pythagorean-leg-square-sum · candidate





## 19. Ordered Leg Squares and Hypotenuse Square

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=20, b=6. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.

View answer and explanation(400, 36, 436)

Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (400, 36, 436).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0019/)

Pythagorean theorem · qf-density13-pythagorean-leg-square-sum · candidate





## 20. Ordered Leg Squares and Hypotenuse Square

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=21, b=7. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.

View answer and explanation(441, 49, 490)

Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (441, 49, 490).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0020/)

Pythagorean theorem · qf-density13-pythagorean-leg-square-sum · candidate





## 21. Ordered Leg Squares and Hypotenuse Square

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=22, b=8. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.

View answer and explanation(484, 64, 548)

Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (484, 64, 548).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0021/)

Pythagorean theorem · qf-density13-pythagorean-leg-square-sum · candidate





## 22. Ordered Leg Squares and Hypotenuse Square

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=23, b=2. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.

View answer and explanation(529, 4, 533)

Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (529, 4, 533).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0022/)

Pythagorean theorem · qf-density13-pythagorean-leg-square-sum · candidate





## 23. Ordered Leg Squares and Hypotenuse Square

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=24, b=3. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.

View answer and explanation(576, 9, 585)

Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (576, 9, 585).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0023/)

Pythagorean theorem · qf-density13-pythagorean-leg-square-sum · candidate





## 24. Ordered Leg Squares and Hypotenuse Square

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=25, b=4. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.

View answer and explanation(625, 16, 641)

Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (625, 16, 641).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0024/)

Pythagorean theorem · qf-density13-pythagorean-leg-square-sum · candidate





## 25. Ordered Leg Squares and Hypotenuse Square

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=26, b=5. Give (a², b², a²+b²), retaining leg order; the last field is the model’s hypotenuse square.

View answer and explanation(676, 25, 701)

Square both positive legs separately and add; this finite side model is not a new universal proof or a completed affine geometry bridge. The exact result, retaining the requested order, is (676, 25, 701).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0025/)

Pythagorean theorem · qf-density13-pythagorean-leg-square-sum · candidate





## 26. Positive Integer Hypotenuse

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=3, b=4, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.

View answer and explanation5

Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0026/)

Pythagorean theorem · qf-density13-pythagorean-hypotenuse-int · candidate





## 27. Positive Integer Hypotenuse

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=6, b=8, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.

View answer and explanation10

Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 10.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0027/)

Pythagorean theorem · qf-density13-pythagorean-hypotenuse-int · candidate





## 28. Positive Integer Hypotenuse

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=9, b=12, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.

View answer and explanation15

Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 15.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0028/)

Pythagorean theorem · qf-density13-pythagorean-hypotenuse-int · candidate





## 29. Positive Integer Hypotenuse

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=12, b=16, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.

View answer and explanation20

Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 20.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0029/)

Pythagorean theorem · qf-density13-pythagorean-hypotenuse-int · candidate





## 30. Positive Integer Hypotenuse

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=15, b=20, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.

View answer and explanation25

Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 25.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0030/)

Pythagorean theorem · qf-density13-pythagorean-hypotenuse-int · candidate





## 31. Positive Integer Hypotenuse

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=18, b=24, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.

View answer and explanation30

Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 30.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0031/)

Pythagorean theorem · qf-density13-pythagorean-hypotenuse-int · candidate





## 32. Positive Integer Hypotenuse

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=21, b=28, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.

View answer and explanation35

Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 35.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0032/)

Pythagorean theorem · qf-density13-pythagorean-hypotenuse-int · candidate





## 33. Positive Integer Hypotenuse

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=24, b=32, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.

View answer and explanation40

Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 40.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0033/)

Pythagorean theorem · qf-density13-pythagorean-hypotenuse-int · candidate





## 34. Positive Integer Hypotenuse

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=27, b=36, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.

View answer and explanation45

Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 45.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0034/)

Pythagorean theorem · qf-density13-pythagorean-hypotenuse-int · candidate





## 35. Positive Integer Hypotenuse

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=30, b=40, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.

View answer and explanation50

Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 50.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0035/)

Pythagorean theorem · qf-density13-pythagorean-hypotenuse-int · candidate





## 36. Positive Integer Hypotenuse

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=33, b=44, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.

View answer and explanation55

Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 55.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0036/)

Pythagorean theorem · qf-density13-pythagorean-hypotenuse-int · candidate





## 37. Positive Integer Hypotenuse

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=36, b=48, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.

View answer and explanation60

Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 60.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0037/)

Pythagorean theorem · qf-density13-pythagorean-hypotenuse-int · candidate





## 38. Positive Integer Hypotenuse

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=39, b=52, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.

View answer and explanation65

Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 65.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0038/)

Pythagorean theorem · qf-density13-pythagorean-hypotenuse-int · candidate





## 39. Positive Integer Hypotenuse

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=42, b=56, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.

View answer and explanation70

Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 70.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0039/)

Pythagorean theorem · qf-density13-pythagorean-hypotenuse-int · candidate





## 40. Positive Integer Hypotenuse

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=45, b=60, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.

View answer and explanation75

Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 75.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0040/)

Pythagorean theorem · qf-density13-pythagorean-hypotenuse-int · candidate





## 41. Positive Integer Hypotenuse

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=48, b=64, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.

View answer and explanation80

Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 80.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0041/)

Pythagorean theorem · qf-density13-pythagorean-hypotenuse-int · candidate





## 42. Positive Integer Hypotenuse

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=51, b=68, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.

View answer and explanation85

Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 85.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0042/)

Pythagorean theorem · qf-density13-pythagorean-hypotenuse-int · candidate





## 43. Positive Integer Hypotenuse

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=54, b=72, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.

View answer and explanation90

Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 90.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0043/)

Pythagorean theorem · qf-density13-pythagorean-hypotenuse-int · candidate





## 44. Positive Integer Hypotenuse

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=57, b=76, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.

View answer and explanation95

Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 95.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0044/)

Pythagorean theorem · qf-density13-pythagorean-hypotenuse-int · candidate





## 45. Positive Integer Hypotenuse

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=60, b=80, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.

View answer and explanation100

Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 100.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0045/)

Pythagorean theorem · qf-density13-pythagorean-hypotenuse-int · candidate





## 46. Positive Integer Hypotenuse

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=63, b=84, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.

View answer and explanation105

Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 105.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0046/)

Pythagorean theorem · qf-density13-pythagorean-hypotenuse-int · candidate





## 47. Positive Integer Hypotenuse

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.

View answer and explanation110

Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 110.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0047/)

Pythagorean theorem · qf-density13-pythagorean-hypotenuse-int · candidate





## 48. Positive Integer Hypotenuse

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=69, b=92, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.

View answer and explanation115

Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 115.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0048/)

Pythagorean theorem · qf-density13-pythagorean-hypotenuse-int · candidate





## 49. Positive Integer Hypotenuse

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=72, b=96, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.

View answer and explanation120

Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 120.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0049/)

Pythagorean theorem · qf-density13-pythagorean-hypotenuse-int · candidate





## 50. Positive Integer Hypotenuse

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=75, b=100, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.

View answer and explanation125

Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 125.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0050/)

Pythagorean theorem · qf-density13-pythagorean-hypotenuse-int · candidate





## 51. Positive Integer Missing Leg

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=3, positive hypotenuse c=5>a; c²−a² is a positive perfect square. Find the other positive leg b.

View answer and explanation4

Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0051/)

Pythagorean theorem · qf-density13-pythagorean-missing-leg-int · candidate





## 52. Positive Integer Missing Leg

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=6, positive hypotenuse c=10>a; c²−a² is a positive perfect square. Find the other positive leg b.

View answer and explanation8

Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 8.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0052/)

Pythagorean theorem · qf-density13-pythagorean-missing-leg-int · candidate





## 53. Positive Integer Missing Leg

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=9, positive hypotenuse c=15>a; c²−a² is a positive perfect square. Find the other positive leg b.

View answer and explanation12

Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 12.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0053/)

Pythagorean theorem · qf-density13-pythagorean-missing-leg-int · candidate





## 54. Positive Integer Missing Leg

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=12, positive hypotenuse c=20>a; c²−a² is a positive perfect square. Find the other positive leg b.

View answer and explanation16

Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 16.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0054/)

Pythagorean theorem · qf-density13-pythagorean-missing-leg-int · candidate





## 55. Positive Integer Missing Leg

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=15, positive hypotenuse c=25>a; c²−a² is a positive perfect square. Find the other positive leg b.

View answer and explanation20

Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 20.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0055/)

Pythagorean theorem · qf-density13-pythagorean-missing-leg-int · candidate





## 56. Positive Integer Missing Leg

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=18, positive hypotenuse c=30>a; c²−a² is a positive perfect square. Find the other positive leg b.

View answer and explanation24

Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 24.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0056/)

Pythagorean theorem · qf-density13-pythagorean-missing-leg-int · candidate





## 57. Positive Integer Missing Leg

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=21, positive hypotenuse c=35>a; c²−a² is a positive perfect square. Find the other positive leg b.

View answer and explanation28

Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 28.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0057/)

Pythagorean theorem · qf-density13-pythagorean-missing-leg-int · candidate





## 58. Positive Integer Missing Leg

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=24, positive hypotenuse c=40>a; c²−a² is a positive perfect square. Find the other positive leg b.

View answer and explanation32

Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 32.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0058/)

Pythagorean theorem · qf-density13-pythagorean-missing-leg-int · candidate





## 59. Positive Integer Missing Leg

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=27, positive hypotenuse c=45>a; c²−a² is a positive perfect square. Find the other positive leg b.

View answer and explanation36

Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 36.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0059/)

Pythagorean theorem · qf-density13-pythagorean-missing-leg-int · candidate





## 60. Positive Integer Missing Leg

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=30, positive hypotenuse c=50>a; c²−a² is a positive perfect square. Find the other positive leg b.

View answer and explanation40

Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 40.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0060/)

Pythagorean theorem · qf-density13-pythagorean-missing-leg-int · candidate





## 61. Positive Integer Missing Leg

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=33, positive hypotenuse c=55>a; c²−a² is a positive perfect square. Find the other positive leg b.

View answer and explanation44

Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 44.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0061/)

Pythagorean theorem · qf-density13-pythagorean-missing-leg-int · candidate





## 62. Positive Integer Missing Leg

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=36, positive hypotenuse c=60>a; c²−a² is a positive perfect square. Find the other positive leg b.

View answer and explanation48

Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 48.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0062/)

Pythagorean theorem · qf-density13-pythagorean-missing-leg-int · candidate





## 63. Positive Integer Missing Leg

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=39, positive hypotenuse c=65>a; c²−a² is a positive perfect square. Find the other positive leg b.

View answer and explanation52

Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 52.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0063/)

Pythagorean theorem · qf-density13-pythagorean-missing-leg-int · candidate





## 64. Positive Integer Missing Leg

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=42, positive hypotenuse c=70>a; c²−a² is a positive perfect square. Find the other positive leg b.

View answer and explanation56

Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 56.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0064/)

Pythagorean theorem · qf-density13-pythagorean-missing-leg-int · candidate





## 65. Positive Integer Missing Leg

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=45, positive hypotenuse c=75>a; c²−a² is a positive perfect square. Find the other positive leg b.

View answer and explanation60

Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 60.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0065/)

Pythagorean theorem · qf-density13-pythagorean-missing-leg-int · candidate





## 66. Positive Integer Missing Leg

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=48, positive hypotenuse c=80>a; c²−a² is a positive perfect square. Find the other positive leg b.

View answer and explanation64

Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 64.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0066/)

Pythagorean theorem · qf-density13-pythagorean-missing-leg-int · candidate





## 67. Positive Integer Missing Leg

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=51, positive hypotenuse c=85>a; c²−a² is a positive perfect square. Find the other positive leg b.

View answer and explanation68

Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 68.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0067/)

Pythagorean theorem · qf-density13-pythagorean-missing-leg-int · candidate





## 68. Positive Integer Missing Leg

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=54, positive hypotenuse c=90>a; c²−a² is a positive perfect square. Find the other positive leg b.

View answer and explanation72

Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 72.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0068/)

Pythagorean theorem · qf-density13-pythagorean-missing-leg-int · candidate





## 69. Positive Integer Missing Leg

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=57, positive hypotenuse c=95>a; c²−a² is a positive perfect square. Find the other positive leg b.

View answer and explanation76

Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 76.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0069/)

Pythagorean theorem · qf-density13-pythagorean-missing-leg-int · candidate





## 70. Positive Integer Missing Leg

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=60, positive hypotenuse c=100>a; c²−a² is a positive perfect square. Find the other positive leg b.

View answer and explanation80

Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 80.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0070/)

Pythagorean theorem · qf-density13-pythagorean-missing-leg-int · candidate





## 71. Positive Integer Missing Leg

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=63, positive hypotenuse c=105>a; c²−a² is a positive perfect square. Find the other positive leg b.

View answer and explanation84

Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 84.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0071/)

Pythagorean theorem · qf-density13-pythagorean-missing-leg-int · candidate





## 72. Positive Integer Missing Leg

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=66, positive hypotenuse c=110>a; c²−a² is a positive perfect square. Find the other positive leg b.

View answer and explanation88

Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 88.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0072/)

Pythagorean theorem · qf-density13-pythagorean-missing-leg-int · candidate





## 73. Positive Integer Missing Leg

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=69, positive hypotenuse c=115>a; c²−a² is a positive perfect square. Find the other positive leg b.

View answer and explanation92

Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 92.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0073/)

Pythagorean theorem · qf-density13-pythagorean-missing-leg-int · candidate





## 74. Positive Integer Missing Leg

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=72, positive hypotenuse c=120>a; c²−a² is a positive perfect square. Find the other positive leg b.

View answer and explanation96

Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 96.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0074/)

Pythagorean theorem · qf-density13-pythagorean-missing-leg-int · candidate





## 75. Positive Integer Missing Leg

Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=75, positive hypotenuse c=125>a; c²−a² is a positive perfect square. Find the other positive leg b.

View answer and explanation100

Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 100.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0075/)

Pythagorean theorem · qf-density13-pythagorean-missing-leg-int · candidate





## 76. Hypotenuse Sign and Square Residual

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=3, b=4, candidate c=-5. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.

View answer and explanation(No, 0, No)

Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (No, 0, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0076/)

Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate





## 77. Hypotenuse Sign and Square Residual

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=3, b=4, candidate c=5. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.

View answer and explanation(Yes, 0, Yes)

Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0077/)

Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate





## 78. Hypotenuse Sign and Square Residual

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=3, b=4, candidate c=0. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.

View answer and explanation(No, -25, No)

Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (No, -25, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0078/)

Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate





## 79. Hypotenuse Sign and Square Residual

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=3, b=4, candidate c=4. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.

View answer and explanation(Yes, -9, No)

Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, -9, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0079/)

Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate





## 80. Hypotenuse Sign and Square Residual

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=3, b=4, candidate c=6. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.

View answer and explanation(Yes, 11, No)

Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, 11, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0080/)

Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate





## 81. Hypotenuse Sign and Square Residual

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=6, b=7, candidate c=-8. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.

View answer and explanation(No, -21, No)

Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (No, -21, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0081/)

Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate





## 82. Hypotenuse Sign and Square Residual

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=7, b=8, candidate c=9. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.

View answer and explanation(Yes, -32, No)

Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, -32, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0082/)

Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate





## 83. Hypotenuse Sign and Square Residual

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=8, b=9, candidate c=-10. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.

View answer and explanation(No, -45, No)

Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (No, -45, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0083/)

Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate





## 84. Hypotenuse Sign and Square Residual

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=9, b=10, candidate c=11. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.

View answer and explanation(Yes, -60, No)

Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, -60, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0084/)

Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate





## 85. Hypotenuse Sign and Square Residual

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=10, b=11, candidate c=-12. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.

View answer and explanation(No, -77, No)

Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (No, -77, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0085/)

Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate





## 86. Hypotenuse Sign and Square Residual

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=11, b=12, candidate c=13. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.

View answer and explanation(Yes, -96, No)

Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, -96, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0086/)

Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate





## 87. Hypotenuse Sign and Square Residual

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=12, b=13, candidate c=-14. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.

View answer and explanation(No, -117, No)

Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (No, -117, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0087/)

Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate





## 88. Hypotenuse Sign and Square Residual

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=13, b=14, candidate c=15. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.

View answer and explanation(Yes, -140, No)

Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, -140, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0088/)

Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate





## 89. Hypotenuse Sign and Square Residual

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=14, b=15, candidate c=-16. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.

View answer and explanation(No, -165, No)

Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (No, -165, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0089/)

Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate





## 90. Hypotenuse Sign and Square Residual

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=15, b=16, candidate c=17. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.

View answer and explanation(Yes, -192, No)

Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, -192, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0090/)

Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate





## 91. Hypotenuse Sign and Square Residual

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=16, b=17, candidate c=-18. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.

View answer and explanation(No, -221, No)

Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (No, -221, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0091/)

Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate





## 92. Hypotenuse Sign and Square Residual

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=17, b=18, candidate c=19. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.

View answer and explanation(Yes, -252, No)

Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, -252, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0092/)

Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate





## 93. Hypotenuse Sign and Square Residual

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=18, b=19, candidate c=-20. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.

View answer and explanation(No, -285, No)

Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (No, -285, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0093/)

Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate





## 94. Hypotenuse Sign and Square Residual

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=19, b=20, candidate c=21. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.

View answer and explanation(Yes, -320, No)

Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, -320, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0094/)

Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate





## 95. Hypotenuse Sign and Square Residual

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=20, b=21, candidate c=-22. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.

View answer and explanation(No, -357, No)

Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (No, -357, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0095/)

Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate





## 96. Hypotenuse Sign and Square Residual

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=21, b=22, candidate c=23. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.

View answer and explanation(Yes, -396, No)

Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, -396, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0096/)

Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate





## 97. Hypotenuse Sign and Square Residual

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=22, b=23, candidate c=-24. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.

View answer and explanation(No, -437, No)

Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (No, -437, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0097/)

Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate





## 98. Hypotenuse Sign and Square Residual

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=23, b=24, candidate c=25. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.

View answer and explanation(Yes, -480, No)

Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, -480, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0098/)

Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate





## 99. Hypotenuse Sign and Square Residual

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=24, b=25, candidate c=-26. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.

View answer and explanation(No, -525, No)

Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (No, -525, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0099/)

Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate





## 100. Hypotenuse Sign and Square Residual

Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=25, b=26, candidate c=27. Give (is c positive?, c²−a²−b², is it positive with zero residual?). c is a candidate only.

View answer and explanation(Yes, -572, No)

Check positivity separately from the squared residual; a nonpositive candidate fails hypotenuse eligibility even with zero residual. The exact result, retaining the requested order, is (Yes, -572, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r01-pythagorean-theorem-0100/)

Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations/validation-report.json)