Question

if a and b are integers, not both of them are zero there exists integers x and y state that (a, b) = ax+ by​

Answer 1

Answer:

Given an integer s and a positive integer g, if there are integers

x and y with x+y=s and (x,y) =g,

then as g divides x and y, it divides their sum, and

so g divides s.

Conversely suppose that we are given that s=n.g for some integer n.

Take x=g and y=(n-1)g

Then clearly x+y = g+(n-1)g = n.g=s,

Also g is a common divisor of x and y.

If d is any common divisor of x and y then d divides

the gcd of x and y i.e.(x,y)=(g,(n-1)g)=g.

So g is indeed the gcd of

x and y, i.e. x+y=s and

(x,y)=g, as desired

Step-by-step explanation:

hope it's helpful to you