High Five, Pythagoras!

A Pythagorean triple is a set of positive integers a<b<ca < b < c such that a2+b2=c2a^2 + b^2 = c^2. Some examples are 32+42=5252+122=13282+152=172.\begin{aligned} 3^2 + 4^2 &= 5^2 \\ 5^2 + 12^2 &= 13^2 \\ 8^2 + 15^2 &= 17^2. \end{aligned} Note that each of these Pythagorean triples contains a multiple of 5.

How many Pythagorean triples {a,b,c}\{a, b, c\} are there for which none of the three numbers is a multiple of 5 and c1000?c \leq 1000?

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