# More fun in 2015, Part 23

Algebra Level 5

How many orthogonal 2 x 2 matrices $$A$$ are there such that all the entries of $$2015A$$ are integers?

For those who don't know matrices yet, we state the problem (less elegantly) in terms of vectors: How many ordered pairs of perpendicular unit vectors $$v,w$$ with two real components are there such that all the components of $$2015v$$ and of $$2015w$$ are integers?

Hint: Find the Pythagorean triples with 2015 as the hypotenuse.

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