  Let a, b be coprime integers. Prove that every integer x > ab - a -b canbe written as na 1 mb wher℃ n,m are non negative ini clicrs. Prov℃ thaiab - a b cannot be expressed in this form.

Question

Abstract Algebra. Please explain everything in detail. help_outlineImage TranscriptioncloseLet a, b be coprime integers. Prove that every integer x > ab - a -b can be written as na 1 mb wher℃ n,m are non negative ini clicrs. Prov℃ thai ab - a b cannot be expressed in this form. fullscreen
Step 1

To prove (or disprove)  the possibilty of required representations under the given conditions

Step 2

We work under the given condtions, a, b positive coprime integers, gcd (a,b)=1. By Euclidean algorithm, we know that ANY integer x can be represented as an+bm . The point is that we require that n and m be non-negative integers. To explain this, consider the example a =10,b=3. The statement says that every integer x > 30-10-3=17 can be represented thus. Note that the integer 13 (less than 17) is also of this form , but 17 cannot be represented thus. We need to prove it in the general case.

Step 3

Describing all integral solutions (no conditions) of the equation an+bm =x. The last line expresses all the solutions (...

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