Finding Positive Integer Solutions to the Equation ( frac{4}{x} - frac{10}{y} 1 )

The problem at hand is to find the number of positive integer solutions for the equation

( frac{4}{x} - frac{10}{y} 1 ).

Solving the Equation

To solve the equation, we begin by rewriting it in a more manageable form. We start by eliminating the denominators by multiplying both sides of the equation by xy:

( xy left( frac{4}{x} - frac{10}{y} right) xy cdot 1 )

This simplifies to:

( 4y - 1 xy )

Rearranging the Equation

Next, we rearrange the equation to:

( xy - 1 - 4y - 1 0 )

By adding 40 to both sides, we get:

( xy - 1 - 4y 39 40 )

We can then factor this equation as follows:

( (y - 10)(x - 4) 40 )

Factor Pairs of 40

To find the positive integer solutions, we need to determine the factor pairs of 40. The positive factor pairs of 40 are:

1, 40 2, 20 4, 10 5, 8 8, 5 10, 4 20, 2 40, 1

For each factor pair (a, b), we have:

( y - 10 a ) and ( x - 4 b )

Substituting these into x and y gives:

( x a 4 ) and ( y b 10 )

Positive Integer Solutions

Factor Pair (a, b)xy (1, 40)550 (2, 20)630 (4, 10)820 (5, 8)918 (8, 5)1215 (10, 4)1414 (20, 2)2412 (40, 1)4411

Therefore, the positive integer solutions to the equation ( frac{4}{x} - frac{10}{y} 1 ) are:

(5, 50) (6, 30) (8, 20) (9, 18) (12, 15) (14, 14) (24, 12) (44, 11)

In total, there are 8 positive integer solutions.

Conclusion

The process of solving the given equation by manipulating the terms, finding the factor pairs of 40, and then solving for the values of x and y leads to the conclusion that there are 8 positive integer solutions.