Purchase Solution

Euclid's Division Lemma and Fundamental Theorem of Arithmetic

Not what you're looking for?

Ask Custom Question

1. Without assuming Theorem 2-1, prove that for each pair of integers j and k (k > 0), there exists some integer q for which j ? qk is positive.
2. The principle of mathematical induction is equivalent to the following statement, called the least-integer principle:
Every non-empty set of positive integers has a least element.
Using the least integer principle, define r to be the least integer for which j ? qk is positive (see Exercise 1). Prove that 0<rk.
3. Use Exercise 2 to give a new proof of Theorem 2?1.
4. Any nonempty set of integers J that fulfills the following two conditions is called an integral ideal:
(i) If n and mare in J, then n+m and n?rn are inJ; and (ii) if n is in J and r is an integer, then rn is in J. Let ) be the set of all integers that are integral multiples of a particular integer rn. Prove that Im is an integral ideal.
5. Prove that every integral ideal J is identical with Jm for some m. [Hint: Prove that if J {O} =, then there exist positive integers in J. By the least-integer principle

Theorem 2-1 is Euclid's Division Lemma : j=qk+r
Please see the attached file for the fully formatted problems.

Attachments
Purchase this Solution

Solution Summary

Euclid's Division Lemma and Fundamental Theorem of Arithmetic are investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

Purchase this Solution


Free BrainMass Quizzes
Probability Quiz

Some questions on probability

Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.