the division algorithm write down complete proof of the division algorithi theorems 27 and 28 in

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The Division Algorithm Write down a complete proof of the Division Algorithm Theorems 27 and 28 in Number Theory 3. The Division Algorithm: Let a be an integer and let b be a natural number. Then there exist unique integers q and r such that a = bq + r and 0 < r < b. Below is an outline of the proof. Proof of existence: Prove Theorem 27 in Number Theory 3. Case 1: b > a (easy case) - This should lead to r = 0. Case 2: b â‰¤ a (note that by Proposition II in Number Theory 2, a = 0) - Consider the set A = {r âˆˆ N | 0 â‰¤ r = a - by for some y âˆˆ Z} Prove A is nonempty. It might be helpful to break it into two cases: Case i: a â‰¥ 0 Case ii: a < 0 Apply the Well-Ordering Principle (see Remark 25 in Number Theory 3) to find the smallest element in A. Prove that 0 < r < b (one way to do this is to show that assuming r â‰¥ b leads to a contradiction). Set q = a - br. Since r is the smallest element in A, let q be an integer such that a - bq < b. Proof of uniqueness: Prove Theorem 28 in Number Theory 3.

          The Division Algorithm
Write down a complete proof of the Division Algorithm Theorems 27 and 28 in Number Theory 3.

The Division Algorithm: Let a be an integer and let b be a natural number. Then there exist unique integers q and r such that a = bq + r and 0 < r < b.

Below is an outline of the proof.

Proof of existence: Prove Theorem 27 in Number Theory 3.

Case 1: b > a (easy case) - This should lead to r = 0.

Case 2: b â‰¤ a (note that by Proposition II in Number Theory 2, a = 0) - Consider the set

A = {r âˆˆ N | 0 â‰¤ r = a - by for some y âˆˆ Z}

Prove A is nonempty. It might be helpful to break it into two cases:

Case i: a â‰¥ 0

Case ii: a < 0

Apply the Well-Ordering Principle (see Remark 25 in Number Theory 3) to find the smallest element in A.

Prove that 0 < r < b (one way to do this is to show that assuming r â‰¥ b leads to a contradiction).

Set q = a - br. Since r is the smallest element in A, let q be an integer such that a - bq < b.

Proof of uniqueness: Prove Theorem 28 in Number Theory 3.

the division algorithm write down complete proof of the division algorithi theorems 27 and 28 in number theory 3 the division algorithm let be an integer and let b natural numnber then there 61525

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