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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