How to Find Positive Integer Solutions to ab 2a 2b

The equation ab 2a 2b might at first seem daunting, but with some algebraic manipulation, it yields insights into its solutions. Let's break it down step-by-step and explore the solutions in detail.

Rearranging the Equation

We start with the given equation:

ab2a 2b

Subtracting 2a 2b from both sides gives us:

ab#8722;2a#8722;2b0

To factorize this, we can add 4 to both sides:

ab#8722;2a#8722;2b 44

Now, factoring the left side, we have:

a(b#8722;2)#8722;2(b#8722;2)4

Which simplifies to:

(a #8722;2)(b #8722;2)4

Now, we need to find pairs where (a - 2)(b - 2) 4. The possible integer pairs are:

(1 - 4) (-2 - 2) (2, 2) (4, 1)

Identifying Positive Integer Solutions

First, consider (a - 2)(b - 2) 4. We need to check which of these lead to positive integer solutions:

CASE 1: a#8722;21#8594;a3#8594;b#8722;24#8594;b6 CASE 2: a#8722;22#8594;a4#8594;b#8722;22#8594;b4 CASE 3: a#8722;24#8594;a6#8594;b#8722;21#8594;b3

The positive integer pairs are (3, 6), (4, 4), and (6, 3).

Exploration of Other Solutions

To explore if there are any other integer values, solving (b 2a / (a - 2)), we find:

For (a, b) (3, 6) and (6, 3), these are valid pairs.

(4, 4) is a straightforward solution from the factored form.

For any large a, b tends to 2, confirming that no higher integer pairs exist.

The equation (b 2a / (a - 2)) also reveals that for large a, (b) tends to 2, showing no higher integer pairs. This leads us to conclude:

3, 6

4, 4

6, 3

are the only positive integer solutions to the equation (ab 2a 2b).

Conclusion

Through algebraic manipulation and logical reasoning, we have shown that the positive integer solutions to the equation (ab 2a 2b) are (3, 6), (4, 4), and (6, 3). Any other integer solutions, if they exist, would not involve both positive integers.