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Right-angled triangles that share a cathetus

Problem 176

Published on Friday, 4th January 2008, 05:00 pm; Solved by 1203; Difficulty rating: 70%

The four right-angled triangles with sides (9,12,15), (12,16,20), (5,12,13) and (12,35,37) all have one of the shorter sides (catheti) equal to 12. It can be shown that no other integer sided right-angled triangle exists with one of the catheti equal to 12.

Find the smallest integer that can be the length of a cathetus of exactly 47547 different integer sided right-angled triangles.