Question

Find the number of ordered pairs of positive integers (x, y) that satisfy the equation (1/x) +(1/y)= 1/2004.

Answer 1

Answer:

This can be rewritten as

1x+1y=1n

1x+1y=1n

1y=1n−1x

1y=1n−1x

y=nxx−n

y=nxx−n

Changing variables to k=x−nk=x−n:

y=n(n+k)k

y=n(n+k)k

y=n2k+n

y=n2k+n

The left side is an integer. What does that tell you about the allowed values of kk, and the corresponding number of valid xx values

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