Proof portfolio II questions. we will prove the Division Algorithm. Theorem 1 (Division Algorithm) If a and b are integers with b21, then there exist unique integers q, r with a=qb+r and 0

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Please solve all three theorem correctly in the order to get positive feedback please show me neat and clean work for it by hand solution needed Please solve three lemna
Proof portfolio II questions.
we will prove the Division Algorithm.
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 1 If a and b are integers with b21, then there exist integers q, r with
a=qb+r and 0<r<b.
Hint: Let S = {a-qb: qe, a-qb20), and apply the well-ordering property.
Lemma 3 The values q, r in (2) are unique.
Hint: Suppose that you have q₁.ri and q2, r2 satisfying (2). Prove that r = r2, and then that g₁ = 92.
2
Transcribed Image Text:Proof portfolio II questions. we will prove the Division Algorithm. 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 1 If a and b are integers with b21, then there exist integers q, r with a=qb+r and 0<r<b. Hint: Let S = {a-qb: qe, a-qb20), and apply the well-ordering property. Lemma 3 The values q, r in (2) are unique. Hint: Suppose that you have q₁.ri and q2, r2 satisfying (2). Prove that r = r2, and then that g₁ = 92. 2
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