Theorem 1 (Division Algorithm). If a and b are integers with b21, then there erist unique integers q, r with a=qb+r and

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Theorem 1 Division Algorithm If A And B Are Integers With B21 Then There Erist Unique Integers Q R With A Qb R And 1 (32.74 KiB) Viewed 45 times

Theorem 1 (Division Algorithm). If a and b are integers with b21, then there erist unique integers q, r with a=qb+r and 0≤r<b. The theorem follows from the next two lemmas. Lemma 3. If a and b are integers with b≥ 1, then there exist integers q, r with a = qb+r and 0≤r<b. Proof. Hint: Let S = {a-qb: qe Z, a-qb20), and apply the well-ordering property. Lemma 5. The values q, r in (4) are unique. Proof. Hint: Suppose that you have q₁, 71 and 92.12 satisfying (4). Prove that r1= r2, and then that qi=92- (4)