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What is the number of integer pairs $(m,n)$ such that $20m-10n=mn$

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1 Answer 1

Iff $20m-10n=mn$, then $$(m+10)(n-20) = mn+10n-20m-200 = -200.$$ If $m,n\in\mathbb{Z}$, then $m+10,n-20\in\mathbb{Z}$, thus the pairs $(m,n)$ can be found by finding $t|-200$ and setting $(m,n) = (t-10,200/t+20)$.

Since $-200 = - 2^3\cdot 5^2$, it has $24$ divisors given by $\pm 2^a\cdot 5^b$, where $a=0,1,2,3$ and $b=0,1,2$.

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