Question

In the following problem, we will work through a proof of an important theorem of arithmetic. Your job will be to read the proof carefully and answer some questions about the argument. Theorem (The Division Algorithm). For any integer n ≥ 0, and for any positive integer m, there exist integers d and r such that n = dm + r and 0 ≤ r < m. Proof: (By strong induction on the variable n.) Let m be an arbitrary positive integer. We will prove by induction on n that for all n ≥ 0 there exist integers d and r such that n = dm + r and 0 ≤ r < m.

2. In the following problem, we will work through a proof of an important theorem of arithmetic. Your job will be to read the proof carefully and answer some questions about the argument. Theorem (The Division Algorithm). For any integer n 2 0, and for any positive integer m, there erist integers d and r such that n - dm +r and 0KT< m Proof: (By strong induction on the variable n.) Let m be an arbitrary positive integer We will prove by induction on n that for all n 2 0 there exist integers d and r such that n=arn r and 0 < r < m, Base Case: The case n- 0. Clearly d-0 and r-0 work, since 0 0-m+0 and r=0<m Induction Hypothesis: Let k 2 0 be a particular integer, and assume that for every integer j with 0 <j S k, there are integers d and r such that j- dm + r and0<r< m Induction Step: Let's consider the integer k +1. If k +1 < m, then we see easily that k +1-Om + (k +1) works (that is, d - 0 and r-k +1). If k +1 2 m, then the integer (k +1) - m is certainly a non-negative integer less than or equal to k, so by the (strong) inductive hypothesis, there are integers d and r such that (k+1) - m-dm + r and 0< r < m. So we get Thus, we have exhibited integers d + 1 and r with the required properties So, for an arbitrary positive integer m, the statement holds for n 0, and whenever it holds for all the non-negative integers less than k +1, it also holds for the integer k +1 By the principle of strong mathematical induction, the division algorithm works for this m we're looking at and all n 2 0. Since the m we started with was arbitrary, the Division Algorithm holds for all n 2 0 and for all m 2 1 (a) Explain how we managed to avoid doing induction on m. (b) Explain why, in the induction step, we needed to treat the case k +1 < m separately from the case k +12m (c) Explain why we needed a strong inductive hypothesis in this argument (d) Show that, in fact, the numbers d and γ are the unique integers with the properties in the statement. That is, let n 2 0 and m 2 1 be integers, and assume that di,rı, d2 and r2 are integers such that also Show that ri r2 (Hint: Show that m divides ri-T2, and use the fact that 0 < rl,T2 < m.) and therefore dı - d2

Answer 1

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