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Master Pythagorean Triples

Learn the secrets of rapid mental computation used in competitive exams. These strategies will help you bypass traditional paper-and-pencil methods.

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Memorize the 5 core primitive triples: (3,4,5), (5,12,13), (8,15,17), (7,24,25), (20,21,29). All others derive from scaling these.

Example

(3,4,5) scaled by 3 → (9,12,15). (5,12,13) scaled by 2 → (10,24,26)

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If you see sides that are multiples of a known triple, the third side scales proportionally.

Example

Legs 9 and 12? That's 3×(3,4,5). Hypotenuse = 3×5 = 15

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For any odd n ≥ 3: the triple is (n, (n²−1)/2, (n²+1)/2).

Example

n=7: (7, 24, 25). n=9: (9, 40, 41). n=11: (11, 60, 61)

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For even n ≥ 4: the triple is (n, (n/2)²−1, (n/2)²+1).

Example

n=4: (3,4,5). n=8: (8,15,17). n=12: (12,35,37)

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The hypotenuse of a primitive triple always ends in 1, 3, 5, 7, or 9 — never an even digit or 0.

Example

(3,4,5)→5, (5,12,13)→3, (8,15,17)→7, (7,24,25)→5

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To instantly verify: compute a² + b² and confirm it equals c². Use known squares to speed this up.

Example

(5,12,13): 25+144=169=13² ✓

📋 Primitive Triples Reference

Primitive triples where a, b, c share no common factor. Sorted by hypotenuse.

a	b	c (hyp)	Verify: a² + b²
3	4	5	9 + 16 = 25
5	12	13	25 + 144 = 169
8	15	17	64 + 225 = 289
7	24	25	49 + 576 = 625
20	21	29	400 + 441 = 841
12	35	37	144 + 1225 = 1369
9	40	41	81 + 1600 = 1681
28	45	53	784 + 2025 = 2809
11	60	61	121 + 3600 = 3721
33	56	65	1089 + 3136 = 4225
16	63	65	256 + 3969 = 4225
13	84	85	169 + 7056 = 7225
36	77	85	1296 + 5929 = 7225
20	99	101	400 + 9801 = 10201