A **Pythagorean triple** is a set of positive integers \(a < b < c\) such that \(a^2 + b^2 = c^2\). Some examples are
\[\begin{align}
3^2 + 4^2 &= 5^2 \\
5^2 + 12^2 &= 13^2 \\
8^2 + 15^2 &= 17^2.
\end{align}\]
Note that each of these Pythagorean triples contains a multiple of 5.

How many Pythagorean triples \(\{a, b, c\}\) are there for which none of the three numbers is a multiple of 5 and \(c \leq 1000?\)

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