find the number of disttinct pairs of postive integers (m,n)such that \(\displaystyle n(m-n)=2009\)

\(\displaystyle 2009=7^2\cdot 41\Longrightarrow\,n\) must be taken from the divisors \(\displaystyle 1,\,7,\,49,\,41,\,287,\,2009\) , and then you must choose \(\displaystyle m\) accordingly.

For example, \(\displaystyle n=1\Longrightarrow\, m=2010\,:\,1(2010-1)=2009\;;\;\;n=7\Longrightarrow\,m=294\,:\) \(\displaystyle 7(294-7)=7\cdot 287=2009\) , etc.

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