# distinct pairs

#### banku12

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

#### tonio

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.

Tonio