Introduction

The problem posed here is to find the pairs of positive integers ( (m, n) ) that satisfy the equation ( frac{1}{m} cdot frac{1}{n} frac{1}{4} ). We will solve this by manipulating the given equation, factoring, and identifying the integer pairs that fulfill the condition.

Solving the Equation

We start with the given equation:

(frac{1}{m} cdot frac{1}{n} frac{1}{4})

Multiplying both sides by (4mn) to clear the denominators, we get:

(4n cdot 4m mn)
This simplifies to:

(16mn mn)

Dividing both sides by (mn) (assuming (mn eq 0)), we have:

(frac{16}{m} 1)
Solving for (mn), we get:

(mn 4m 4n)
Then, rearranging the equation:

(mn - 4m - 4n 16 16)

This can be factored as:

((m - 4)(n - 4) 16)

We now need to find all pairs of integers that multiply to 16 and are both positive.

Identifying the Factor Pairs

The factor pairs of 16 are:

(1 times 16)(2 times 8)(4 times 4)(8 times 2)(16 times 1)

We will consider each pair to determine the corresponding values of (m) and (n).

Determining the Integer Pairs

From the factor pairs, we have:

( (m - 4, n - 4) (1, 16) ): ( n - 4 16 Rightarrow n 20 ) ( m - 4 1 Rightarrow m 5 ) Thus, one pair is ( (5, 20) ).( (m - 4, n - 4) (2, 8) ): ( n - 4 8 Rightarrow n 12 ) ( m - 4 2 Rightarrow m 6 ) Thus, another pair is ( (6, 12) ).( (m - 4, n - 4) (4, 4) ): ( n - 4 4 Rightarrow n 8 ) ( m - 4 4 Rightarrow m 8 ) Thus, another pair is ( (8, 8) ).( (m - 4, n - 4) (8, 2) ): ( n - 4 2 Rightarrow n 5 ) ( m - 4 8 Rightarrow m 12 ) Thus, another pair is ( (12, 5) ).( (m - 4, n - 4) (16, 1) ): ( n - 4 1 Rightarrow n 5 ) ( m - 4 16 Rightarrow m 20 ) Thus, the last pair is ( (20, 5) ).

The valid pairs of positive integers that satisfy the equation are:

((5, 20)), ((20, 5)), ((6, 12)), ((12, 6)), ((8, 8))

Conclusion

Counting these pairs, we find that there are 5 pairs of positive integers that satisfy the equation. Therefore, the final answer is:

(boxed{5})