Solving Integer Pairs for the Equation (mn^2 - 5mn k^2)

Understanding the problem at hand, we need to find all positive integer pairs (m) and (n) that satisfy the given equation (mn^2 - 5mn k^2), where (k) is also an integer. This equation simplifies to:

1. Identifying Constraints on (k)

First, let's consider the constraints on (k). We need:

(k leq mn) (k geq mn - 3)

Given these conditions, (k) can only take on two specific values in the positive integer domain. These values are:

(k mn - 1) with (n 3m - 1) (k mn - 2) with (m 3n - 4)

2. Calculating the Number of Integer Pairs

Let's now calculate the number of pairs ((m, n)) that satisfy these conditions and where (mn leq 100).

3. Case 1: (k mn - 1) with (n 3m - 1)

In this case, we need to find pairs ((m, n)) such that:

(3m - 1 leq 100)

Therefore, (m leq 33), giving us 33 pairs.

4. Case 2: (k mn - 2) with (m 3n - 4)

In this case, we need to find pairs ((m, n)) such that:

(3n - 4 leq 100)

Therefore, (n leq 31), giving us 31 pairs.

Combining both cases, we have a total of 64 pairs that satisfy the given equation within the constraints.

3. Simple Derivation and Verification

The process involves some trial and verification. We decompose the problem into manageable pieces and derive possible solutions. For instance, consider the equation:

(mn^2 - 5mn k^2)

This can be rewritten as:

(mn^2 - k^2 5mn)

Then we can further decompose it into:

( [k, m, n] cdot [k - m, n] - mn 4m )

This implies we need to choose (m) and (n) such that the left-hand side is a multiple of 4.

4. Specific Solutions

For the first type of pair ((m, n 3m - 1)), we get:

(m leq 33) giving us 33 pairs.

For the second type of pair ((m 3n - 4, n)), we get:

(n leq 31) giving us 31 pairs.

Combine these results to get the total count.

4. Conclusion

Through a series of logical steps and calculations, we determined that there are a total of 64 pairs of ((m, n)) that satisfy the equation (mn^2 - 5mn k^2), where (k) is an integer and (mn leq 100).