Originally Posted by

**watzer** Hi! I'm stuck!

How to prove that a equation x + y = n always have a positive solution(x,y >= 0) when n > z?

What is z??

"a Linear Diophantine equation"

(Ex: 69x + 39y = n, when n > 20)

This is false: any solution to this equation will have x or y negative if , for example, n = 21

Tonio

My steps have been:

Solve the equation as u solve diofantic equations:

so u get:

Ex:

x + y = 1 (in my case GCD(x,y) were one)

then i did multiply all solutions with the value EX: 21,22,23,24,25 ( if n > 20 )

and i got diffrent x,y values on all five wich had all one value of K were X,Y where positive.

(ex:x = -110-49k, y = 155 + 69k)

and all solutions over n > z had atleast one solution wich both x and y were positive.

And then i got a pattern similar to this

Ex:

n 21---22----23---24----25

X 50---52----54---56----58

Y 100--105--110--115--120

K 2-----2-----3-----3------4

But have no idea how to prove it with induction.

Is it posible to prove X then prove Y and by that K is proven to consist and always have a solution ?