Chapter 4: Elementary Number Theory and Methods of Proof Discrete Structures Chapter 4: Elementary Number Theory and Methods of Proof 4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem Be especially critical of any statement following the word “obviously”. – Anna Pell Wheeler, 1883-1966 4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem

Theorem 4.4.1 – The Quotient-Remainder Theorem Given any integer n and positive integer d,  unique integers q and r s.t. n = dq + r and 0  r < d 4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem

Definitions Given an integer n and a positive integer d, n div d = the integer quotient obtained when n is divided by d. n mod d = the nonnegative integer remainder obtained when n is divided by d. Symbolically, if n and d are integers and d > 0, then n div d = q and n mod d = r  n = dq + r Where q and r are integers and 0  r < d. 4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem

Example – pg. 189 # 8 & 9 Evaluate the expressions. a. 50 div 7 b. 50 mod 7 a. 28 div 5 b. 28 mod 5 4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem

Theorems Theorem 4.4.2 – The Parity Property Any two consecutive integers have opposite parity. Theorem 4.4.3 The square of any odd integer has the form 8m + 1 for some integer m. 4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem

Definition For any real number x, the absolute value of x, is denoted |x|, is defined as follows: 4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem

Lemmas Lemma 4.4.4 For all real numbers r, -|r|  r  |r| Lemma 4.4.5 Lemma 4.4.6 – The Triangle Inequality For all real numbers x and y, |x + y|  |x| + |y| 4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem

Example – pg. 189 # 24 Prove that for all integers m and n, if m mod 5 = 2 and n mod 5 = 1, then mn mod 5 = 2. 4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem

Example – pg. 190 #36 Prove the following statement. The product of any four consecutive integers is divisible by 8. 4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem

Example – pg. 190 #42 Prove the following statement. Every prime number except 2 and 3 has the form 6q + 1 or 6q + 5 for some integer q. 4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem

Example – pg. 190 #45 Prove the following statement. For all real numbers r and c with c  0, if -c  r  c, then |r|  c. 4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem

Example – pg. 190 # 49 If m, n, and d are integers, d > 0, and m mod d = n mod d, does it necessarily follow that m = n? That m – n is divisible by d? Prove your answers. 4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem