menu
bartleby
search
close search
Hit Return to see all results

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.

Let 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.
help_outline

Image Transcriptionclose

Let 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
check_circleAnswer
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.

fullscreen
Step 3

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

fullscreen

Want to see the full answer?

See Solution

Check out a sample Q&A here.

Want to see this answer and more?

Our solutions are written by experts, many with advanced degrees, and available 24/7

See Solution
Tagged in

Math

Advanced Math

Related Advanced Math Q&A

Find answers to questions asked by student like you

Show more Q&A add
question_answer

Q: I need help for the proof of question 2. Thank you very much!

A: Given: 

question_answer

Q: I only need the explanation of number 8. I do not know how to do it. Thanks

A: As per norms, the subparts for question 8 are answered.To analyze the given relation f. 

question_answer

Q: what is 2+ 2?

A: Obtain the value of the expression as follows.The number 2 is will be located at a distance of 2 uni...

question_answer

Q: Subject: Elements if Set Theory By Herbert B. Enderton   Having trouble with a written assignment gi...

A: To establish the formula for the sum of the first n squares, using induction.

question_answer

Q: 5. Let (n) be a uniformly bounded sequence of functions which are Riemann integrable on [a, b, and p...

A: By the hypothesis there exists an real number M such that Supremum of x belongs to [a,b] of |fn(x)|&...

question_answer

Q: the book is "First Course on Fuzzy Theory and Applications"

A: to check if the given fuzzy sets satisfy the convexity condition

question_answer

Q: please explain a and b to me step by step and dont skip any steps. pls write clearly and  i will rat...

A: (a)Consider the given matrix:

question_answer

Q: What is a formula to calculate the sum of the first n positive integers? Could you explain why this ...

A: Obtain the sum of first n positive integers as follows.The general formula to calculate the sum of f...

question_answer

Q: Q1- Prove that ∼H is an equivalence relation on G. That is, ∼H satisfies: (i) For every g ∈ G, g ∼H...

A: As per norms, the first three sub-parts of Question 1 are answered.  This concerns proving that the ...

Sorry about that. What wasn’t helpful?