Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question

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 $b=0,1,2$.

share|cite|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.