How to find the number of solutions to the equation $\frac {50}{x} + \frac {126}{y} = 1$ , assuming that x and y are positive, integer numbers?

Question

Find the number of solutions to the equation $\frac {50}{x} + \frac {126}{y} = 1$ , assuming that both $x$ and $y$ are positive integers. How should I approach this problem?

Answer 1

HINT: write $$x=\frac{50y}{y-126}$$ and this can be written as $$x=\frac{6300}{y-126}+50$$

Answer 2

Clearing fractions you have $$50y+126x=xy$$ which can be rewritten $$(x-50)(y-126)=6300$$

For every integer factorisation $ab=6300$ you have a solution with $x=a+50$. Count with care.

Answer 3

There are 54 possible pairs (x,y) satisfying your equation. Multiply your equation by xy to get 50y + 126x = xy. This equation is equivalent to (x-50)(y-126)= 6300. Since there are 54 positive divisors of 6300 and for each integer factorization of 6300 = ab we get x=50+a, y=126+b there are a total of 54 possible solutions.