If \(a\) and \(b\) are positive integers from \(1\) to \(999\) [inclusive], how many ordered pairs \((a, b)\) are there such that \[(a+36b)(b+36a)\] is an integral power of \(2\)?

This problem is a reworded version of a problem that appeared at the APMO.

This problem is a part of the set "Olympiads and Contests Around the World - 2". You can see rest of the problems here.

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