Big circle, little circles

Number Theory Level 4

\(r\), \(N\) and \(R\) are nonzero positive integers, and \(N\) > 1.

\(r\) is the radius of \(N\) circles whose areas together are equal to the area of a single circle with radius \(R\).

For example, if \(R = 100\), \(r\) could be 1, and there would be \(N=10\,000\) circles. \(r\) could also be 2, and there would be \(N=2500\) circles. However, \(r\) could not be 3, since there is no way to divide the big circle evenly into little circles with that radius.

How many different possible values of the small radius \(r\) exist for a large radius of \(R = 90\,000\)?


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