How can I solve an equation with two variables where x is maximum?

Lets say I have an equation - x^2+y^2=100 - obviously开发者_JAVA百科 there's more than one solution.

I want to make Mathematica 8 give me the solution (where only natural numbers involved) where x will be maximized (i.e x=10, y=0)

I'm pretty new to Mathematica - and got really confused with whats going on...

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Without the Diophantine explicit requierment:

Maximize[{x , x^2 + y^2 == 100}, {x, y}]
(*
-> {10, {x -> 10, y -> 0}}
*)

Edit

As you can see, the result is a two elements list. The first element (10) is the value for x (the function for which the maximization is performed). The second element is {x -> 10, y -> 0}, corresponding to the assignment rules for the variables at the max point.

Note that here we are maximizing x, so the value 10 is repeated in both elements, but that is not always the case, as we usually want to maximize a general function of the variables, and not the vars themselves.

In this particular case, we have two straightforward ways to assign the max value of x to n:

Using the first element of the list:

n = First@Maximize[{x , x^2 + y^2 == 100}, {x, y}]  

Or more general, using the appropriate rule:

n = x /. Last@Maximize[{x, x^2 + y^2 == 100}, {x, y}]