Theorem 1 (Division Algorithm): If a and b are integers with b≥ 1, then there exist unique integers q, r with a = qb+r a

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Theorem 1 Division Algorithm If A And B Are Integers With B 1 Then There Exist Unique Integers Q R With A Qb R A 1 (35.17 KiB) Viewed 22 times

Theorem 1 (Division Algorithm): If a and b are integers with b≥ 1, then there exist 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. Hint: Let S = {a - qb: q € Z, a- qb>0}, and apply the well-ordering property. Proof. Lemma 5: The values q, r in (4) are unique. Hint: Suppose that you have q₁, 7₁ and 92,72 satisfying (4). Prove that r₁ = 72, and then that qi=92. Proof. (4)