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Understanding Division: Definitions, Properties, and the Division Algorithm Explained

This article provides a comprehensive overview of division, starting with essential definitions such as divisors, multiples, and prime versus composite numbers. We explore the Division Algorithm, which states that for any integers (a) and (b) (where (b>0)), there exist unique integers (q) and (r) such that (a = bq + r) (where (0 leq r < b)). The proof of existence and uniqueness for the quotient and remainder is discussed, illustrating how to divide integers. Examples clarify these concepts for positive and negative integers.

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Understanding Division: Definitions, Properties, and the Division Algorithm Explained

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  1. All About Division

  2. Definition • A nonzero integer t is a divisor of an integer s if there is an integer u such that s = tu. • If t is a divisor of s, we write t | s, read 't divides s'

  3. Vocabulary • We also say that s is a multiple of t. • A prime is an integer greater than 1 whose only positive divisors are 1 and itself. • A positive integer with divisors other than itself and 1 is composite.

  4. Example • 8|24 because 24 = 8*3 8 is a divisor of 24. 24 is a multiple of 8. 24 is not prime. 24 is composite.

  5. Theorem 0.1 Division Algorithm • Let a and b be integers with b > 0. • There exist unique integers q and r with the property that a = bq + r, where 0 ≤ r < b

  6. a qb (q+1)b My Proof (Existence) • Consider every multiple of b. • Since a is an integer, it must lie in some interval [qb,(q+1)b). • Set r = a – qb. • Note that r is an integer with 0 ≤ r < b and a = qb + r as required.

  7. Proof: (Uniqueness) • Suppose a = q1b + r1and a = q2b + r2where 0 ≤ r1,r2 < b. We may suppose that r1 ≥ r2. • Then 0 ≤ r1 – r2 < b, But r1 – r2 = (a – q1b ) – (a – q2b) = (q2 – q1)b • Hence r1 – r2 is a non-negative multiple of b that is strictly less than b. • It follows that r1 – r2 = 0. • So r1 = r2and then q1 = q2 as required.

  8. Note • In our existence proof, we said that "a must lie" in an interval … • How do we know that? It is not a consequence of algebra(+ - * \) or of order ( < = > ) • It is a consequence of the well ordering principle.

  9. a–bk r=a–bq a Gallian's proof • S = {a–bk | k is an integer, a–bk > 0} • By the WOP, there is a smallest element of S, call it r.

  10. It remains to show that r < b. To get a contradiction, suppose r ≥ b. Then r – b ≥ 0. But r = a – bk for some integer k, So r – b = a – (k+1)b ≥ 0 Then r – b is in S and smaller than r. This contradiction shows r < b.

  11. Given a, b find q, r • Divide 38 by 7: • Write: 38 = 5*7 + 3, so q = 5, r = 3 • Divide -38 by 7: • Write: -38 = -6*7 + 4, so q = -6, r = 4

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