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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 explanation
5
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
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 explanation
10
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
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 explanation
15
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
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 explanation
20
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
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 explanation
25
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
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 explanation
30
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
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 explanation
35
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
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 explanation
40
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
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 explanation
45
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
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 explanation
50
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
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 explanation
55
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
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 explanation
60
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
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 explanation
65
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
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 explanation
70
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
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 explanation
75
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
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 explanation
80
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
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 explanation
85
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
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 explanation
90
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
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 explanation
95
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
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 explanation
100
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
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 explanation
105
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
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 explanation
110
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
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 explanation
115
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
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 explanation
120
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
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 explanation
125
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
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 explanation
4
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
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 explanation
8
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
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 explanation
12
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
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 explanation
16
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
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 explanation
20
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
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 explanation
24
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
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 explanation
28
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
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 explanation
32
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
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 explanation
36
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
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 explanation
40
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
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 explanation
44
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
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 explanation
48
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
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 explanation
52
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
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 explanation
56
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
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 explanation
60
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
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 explanation
64
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
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 explanation
68
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
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 explanation
72
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
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 explanation
76
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
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 explanation
80
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
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 explanation
84
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
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 explanation
88
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
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 explanation
92
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
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 explanation
96
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
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 explanation
100
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Pythagorean theorem · qf-density13-pythagorean-candidate-residual · candidate
Canonical question JSONL · English JSONL · Validation report