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Euclid's Division Lemma and Fundamental Theorem of Arithmetic

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
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