solution of 1/x+1/y=1/n

Theorem 1.

Given an integer n, if there exist integers x and y such that


then one has

x = n(u+v)u
y = n(u+v)v

where u and v are integers such that uv divides n.


To begin, cross multiply to obtain


Since this involves setting a productPlanetmathPlanetmathPlanetmath equal to another product, we can think in terms of factorization. To clarify things, let us pull out a common factor and write x=kv and y=ku, where k is the greatest common factor and u is relatively prime to v. Then, cancelling a common factor of k, our equation becomes the following:


This is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to


Since u and v are relatively prime, it follows that u is relatively prime to u+v and that v is relatively prime to u+v as well. Hence, we must have that uv divides n,

Now we can obtain the general solution to the equation. Write n=muv with u and v relatively prime. Then, substituting into our equation and cancelling a u and a v, we obtain


so the solution to the original equation is

x = mv(u+v)
y = mu(u+v)

Using the definition of m, this can be rewritten as

x = n(u+v)u
y = n(u+v)v.

Title solution of 1/x+1/y=1/n
Canonical name SolutionOf1x1y1n
Date of creation 2013-03-22 16:30:44
Last modified on 2013-03-22 16:30:44
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 13
Author rspuzio (6075)
Entry type Theorem
Classification msc 11D99