Who Knows Pythagoras?

Number Theory Level 5

\(\bullet\) Let \(a\) be the hypotenuse of a right angled triangle; with other two sides being \(5\) and \(12\) respectively.

\(\bullet\) Let \(b\) be the sum of all positive integers which cannot be the sides of a integer-sided right angled triangle.

\(\bullet\) Let \(c\) be the greatest common divisor of all possible values of \(xyz\), where \(x\), \(y\), and \(z\) are the integral sides of a right angled triangle subject to \(x+y+z \ge 100\)

\(\bullet\) Let \(d\) be the number of different integer-sided right-angled triangles with one of the sides equal to \(385\).

Determine \(a+b+c+d\).


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