Competition problem — Number theory

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Number theory Difficulty 5.5 AIME, harder Find the answer

2. Find all Shang Gao triangles, and all primitive Shang Gao triangles, with one side length of (i) 15, (ii) 22, (iii) 50.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

2. (i) Primitive: $\{15,8,17\},\{15,112,113\}$ , Non-primitive: $\{15,36,39\}$ , $\{15,20,25\},\{9,12,15\}$ ; (ii) No primitive, Non-primitive: $\{22,120,122\}$ , (iii) No primitive, Non-primitive: $\{14,48,50\},\{30,40,50\},\{50,120,130\},\{50,624,626\}$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.