Solving the Equation for Ordered Pairs of Integers

The problem at hand is to determine the number of ordered pairs of integers (x, y) that satisfy the equation 1 / x1 / y 1 / 2004. To solve this problem, we need to first simplify and analyze the given equation.

Starting with the given equation:

1 / x1 / y 1 / 2004

We can rewrite this using the property of fractions:

1 / (xy) 1 / 2004

This implies that:

xy 2004

The next step is to factorize 2004. The prime factorization of 2004 is:

2004 2^2 * 3^1 * 167^1

According to the theorem for finding the number of factors, the number of positive integer factors of N is given by multiplying the incremented exponents of its prime factorization. For 2004:

N_{factors} (2 1)(1 1)(1 1) 3 * 2 * 2 12

This means there are 12 positive integer pairs for which xy 2004. However, since the problem is about ordered pairs of integers, we need to consider both positive and negative factors, which doubles the count. Hence, the total number of ordered pairs is:

2 * 12 24

However, we need to exclude the case where either x or y is zero, as these would not satisfy the original equation. The only scenario where xy 0 is not a solution is when d -k. As we determined earlier, the only value of d that leads to such a solution is d -k. Since d -k is not a solution, we subtract one from the total count:

24 - 1 23

Therefore, the number of ordered pairs of integers that satisfy the equation 1 / x1 / y 1 / 2004 is 23.

By following these steps and using the properties of integer factorization, we can systematically solve such problems and arrive at a definitive answer.