Solving the Equation 5a - ab 9b^2 for Positive Integers a and b

The given equation is 5a - ab 9b^2, where both a and b are positive integers. We aim to find all possible solutions for (a, b) that satisfy this equation. There are various methods to approach this problem, and we will discuss both a brute force method and a manual approach.

Brute Force Approach

Using a programming language like J, a brute force approach can be implemented to find the solutions. Here's an example of a brute force solution using the J programming language:

a b : .n : .quot.;odo 210000
{n~9b^25a-ab

The results from the brute force algorithm indicate that there are two solutions:

|12 2│144 4│

Thus, the solutions are (a, b) (12, 2) and (144, 4).

Manual Approach

The equation 5a - ab 9b^2 can be treated as a quadratic equation in terms of b. By rearranging the terms, we get:

b frac{-a pm sqrt{a^2 - 180}}{18}

However, this only provides a general form, and for integer solutions, further brute force or manual checking is necessary. We can solve this equation by substituting possible values of b and checking if a remains a positive integer.

From here, let's try b from 1 to 4 and see what values make a an integer:

b a Integer Check 1 frac{9}{4} No 2 12 Yes 3 frac{81}{2} No 4 144 Yes

Therefore, the positive integer solutions for (a, b) are (12, 2) and (144, 4).

Further Simplification

Let's simplify the equation to further validate the solutions:

5a - ab 9b^2

A rearranged form is:

a frac{9b^2}{5 - b}

This equation implies that for a to be a positive integer, the denominator (5 - b) must be a divisor of 9b^2 and (5 - b) 0. The only positive integer values for b that satisfy these conditions are b 2 and b 4. Let's substitute these values:

For b 2, a frac{9 times 4}{5 - 2} 12 For b 4, a frac{9 times 16}{5 - 4} 144

Hence, the positive integer solutions are (a, b) (12, 2) and (144, 4).

Conclusion

In conclusion, the equation 5a - ab 9b^2 has two positive integer solutions: (12, 2) and (144, 4). This can be determined using both brute force and manual methods, confirming the correctness of these solutions.