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ax + by = 1

The extension states that, if a and b are coprime the least natural number k for which all natural numbers greater than k can be expressed in the form:

ax + by

Is a+b-1. (Where x and y are natural numbers)

I am trying to prove this extension. I have used C++ to test it for a few specific examples, but this does not cut the mustard. I need a rigorous proof. I looked up ideals and ring theory, but I am at a loss because I am doing a physics and applied maths degree, I have very little knowledge of pure mathematics. Any help would be greatly appreciated, thanks in advance.