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Problem 176

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Right-angled triangles that share a cathetus

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.

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拥有等长直角边的直角三角形

如下四个直角三角形，三边长分别为(9,12,15)，(12,16,20)，(5,12,13)和(12,35,37)，均有一条直角边长为12。可以证明不存在其它整数边长直角三角形拥有一条长为12的直角边。

找出使得恰好有47547个不同的整数边长直角三角形拥有该长度直角边的最小整数。

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