Chapter 4: Elementary Number Theory and Methods of Proof 4.4 Direct Proof and Counter Example IV:...
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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 1 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
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Transcript of Chapter 4: Elementary Number Theory and Methods of Proof 4.4 Direct Proof and Counter Example IV:...
4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem
1
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
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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
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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
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Example – pg. 189 # 8 & 9
• Evaluate the expressions.
8. a. 50 div 7 b. 50 mod 7
9. a. 28 div 5 b. 28 mod 5
4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem
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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
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Definition
• For any real number x, the absolute value of x, is denoted |x|, is defined as follows:
if 0
if 0
x xx
x x
4.4 Direct Proof and Counter Example IV: Division into Cases and the Quotient Remainder Theorem
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Lemmas
• Lemma 4.4.4
For all real numbers r, -|r| r |r|
• Lemma 4.4.5
For all real numbers r, |-r| = |r|
• 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
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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
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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
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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
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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
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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.
