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**Quotient-Remainder Theory, Div and Mod**

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. Quotient: q=𝑛 𝑑𝑖𝑣 𝑑= 𝑛 𝑑 Reminder: r=𝑛 𝑚𝑜𝑑 𝑑=𝑛−𝑑𝑞

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Exercise Prove that for all integers a and b, if a mod 7 = 5 and b mod 7 = 6 then ab mod 7 = 2. What values are 𝑑, 𝑞, and 𝑟?

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Exercise Prove that for all integers a and b, if a mod 7 = 5 and b mod 7 = 6 then ab mod 7 = 2. Hint: 𝑑 = 7 𝑎 = 7𝑚+5, b = 7𝑛+6 𝑎𝑏 = 7𝑚+5 7𝑛+6 =49𝑚𝑛+42𝑚+35𝑛+30=7 7𝑚𝑛+6𝑚+3𝑛+4 +2

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**Floor & Ceiling Floor: If 𝑥 is a real number and 𝑛 is an integer, then**

Definition: Floor: If 𝑥 is a real number and 𝑛 is an integer, then 𝑥 =𝑛 ⟺ 𝑛≤𝑥<𝑛+1 Ceiling: if 𝑥 is a real number and 𝑛 is an integer, then 𝑥 =𝑛 ⟺ 𝑛−1<𝑥≤𝑛 𝑥 𝑛 𝑛+1 floor of 𝑥= 𝑥 𝑥 𝑛−1 𝑛 ceiling of 𝑥= 𝑥

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**Relations between Proof by Contradiction and Proof by Contraposition**

To prove a statement ∀𝑥 in 𝐷, if 𝑃 𝑥 then 𝑄(𝑥) Proof by contraposition proves the statement by giving a direct proof of the equivalent statement ∀𝑥 in 𝐷, if 𝑄 𝑥 is false then 𝑃 𝑥 is false Proof by contradiction proves the statement by showing that the negation of the statement leads logically to a contradiction. Suppose 𝑥 is an arbitrary element of 𝐷 such that ~𝑄(𝑥) ~𝑃(𝑥) Sequence of steps Suppose ∃𝑥 in 𝐷 such that 𝑃(𝑥) and ~𝑄(𝑥) Same sequence of steps Contradiction: 𝑃(𝑥) and ~𝑃(𝑥)

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**Summary of Chapter 4 Number theories: Proofs:**

Even, odd, prime, and composite Rational, divisibility, and quotient-remainder theorem Floor and ceiling The irrationality of 2 and gcd Proofs: Direct proof and counterexample Indirect proof by contradiction and contraposition

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**𝑟 is rational ⇔ ∃ integers 𝑎 and 𝑏 such that 𝑟 = 𝑎 𝑏 and 𝑏 ≠ 0.**

The Irrationality of 2 How to proof: Direct proof? Proof by contradiction? Proof by contraposition? If 𝑟 is a real number, then 𝑟 is rational ⇔ ∃ integers 𝑎 and 𝑏 such that 𝑟 = 𝑎 𝑏 and 𝑏 ≠ 0. A real number that is not rational is irrational.

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**The Irrationality of 2 Proof by contradiction: Starting point:**

Negation: 2 is rational. To show: A contradiction. 2 = 𝑚 𝑛 , where 𝑚 and 𝑛 are integers with no common factors and 𝑛≠0, by definition of rational. 𝑚 2 = 2 𝑛 2 , by squaring and multiplying both sides with 𝑛 2 𝑚 2 is even, then 𝑚 is even. Let 𝑚 = 2𝑘 for some integer 𝑘. (2𝑘) 2 =4 𝑘 2 =2 𝑛 2 , by substituting 𝑚 = 2𝑘 into 𝑚 2 = 2 𝑛 2 . 𝑛 2 is even, and so 𝑛 is even. Hence both 𝑚 and 𝑛 have a common factor of 2.

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**Irrationality of 1+3 2 Proof by contradiction: Starting point:**

Negation: is rational. To show: A contradiction.

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**1+3 2 = 𝑎 𝑏 for some integers 𝑎 and 𝑏 with 𝑏≠0.**

Irrationality of Proof by contradiction: By definition of rational, = 𝑎 𝑏 for some integers 𝑎 and 𝑏 with 𝑏≠0. It follows that 3 2 = 𝑎 𝑏 − by subtracting 1 from both sides = 𝑎 𝑏 − 𝑏 𝑏 by substitution = 𝑎−𝑏 𝑏 by the rule for subtracting fractions with a common denominator Hence, 2 = 𝑎−𝑏 3𝑏 by dividing both sides by 3. 𝑎 − 𝑏 and 3𝑏 are integers and 3𝑏≠0 by the zero product property. Hence 2 is quotient of the two integers 𝑎 − 𝑏 and 3𝑏 with 3𝑏≠0, so 2 is rational by the definition of rational. This contradicts the fact that 2 is irrational.

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**Property of a Prime Divisor**

Proposition 4.7.3 For any integer 𝑎 and any prime number 𝑝, if 𝑝|𝑎 then 𝑝 | (𝑎+1) If a prime number divides an integer, then it does not divide the next successive integer. Starting point: there exists an integer 𝑎 and a prime number 𝑝 such that 𝑝 | 𝑎 and 𝑝 | (𝑎 + 1). To show: a contradiction. 𝑎 = 𝑝𝑟 and 𝑎 + 1 = 𝑝𝑠 for some integers 𝑟 and 𝑠 by definition of divisibility. It follows that 1 = (𝑎 + 1) − 𝑎 = 𝑝𝑠 − 𝑝𝑟 = 𝑝(𝑠 − 𝑟 ), 𝑠 − 𝑟 =1/𝑝, by dividing both sides with 𝑝. 𝑝 > 1 because 𝑝 is prime, hence, 1/𝑝 is not an integer, thus 𝑠−𝑟 is not an integer, which is a contradict 𝑠−𝑟 is an integer since 𝑟 and 𝑠 are integers. if 𝑛 and 𝑑 are integers and 𝑑≠0: 𝑑|𝑛 ⟺ ∃ an integer 𝑘 such that 𝑛=𝑑∙𝑘

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**Infinitude of the Primes**

Theorem Infinitude of the Primes The set of prime numbers is infinite. Proof by contradiction: Starting point: the set of prime number is finite. To show: a contradiction. Assume a prime number 𝑝 is the largest of all the prime numbers 2, 3, 5, 7, 11,. . . ,𝑝. Let 𝑁 be the product of all the prime numbers plus 1: 𝑁 = (2·3·5·7·11· · ·𝑝) + 1 Then 𝑁 > 1, and so, by Theorem (any integer larger than 1 is divisible by a prime number) , 𝑁 is divisible by some prime number q. Because q is prime, q must equal one of the prime numbers 2, 3, 5, 7, 11, , 𝑝. Thus, by definition of divisibility, 𝑞 divides 2·3·5·7·11· · ·𝑝, and so, by Proposition 4.7.3, 𝑞 does not divide (2·3·5·7·11· · ·𝑝) + 1, which equals 𝑁. Hence N is divisible by q and N is not divisible by q, and we have reached a contradiction. [Therefore, the supposition is false and the theorem is true.]

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**Greatest Common Divisor (GCD)**

The greatest common divisor of two integers a and b is the largest integer that divides both 𝑎 and 𝑏. Exercise: gcd 72,63 =9, since 72=9∙8 and 63=9∙7 gcd , = 2 20 , since = 2 20 ∙ 5 20 and = 2 30 ∙ 3 30 Definition Let 𝑎 and 𝑏 be integers that are not both zero. The greatest common divisor of 𝑎 and 𝑏, denoted gcd(a, b), is that integer 𝑑 with the following properties: 𝑑 is common divisor of both a and b, in other words, 𝑑 | 𝑎, and 𝑑 | 𝑏. For all integers 𝑐, if 𝑐 is a common divisor of both 𝑎 and 𝑏, then 𝑐 is less than or equal to 𝑑. In other words, for all integers 𝑐, if 𝑐 | 𝑎, and 𝑐 |𝑏, then c≤𝑑.

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**Greatest Common Divisor (GCD)**

Lemma 4.8.1 If 𝑟 is a positive integer, then gcd(𝑟,0) = 𝑟. Proof: Suppose 𝑟 is a positive integer. [We must show that the greatest common divisor of both 𝑟 and 0 is 𝑟.] 𝑟 is a common divisor of both 𝑟 and 0 because r divides itself and also 𝑟 divides 0 (since every positive integer divides 0). No integer larger than 𝑟 can be a common divisor of 𝑟 and 0 (since no integer larger than 𝑟 can divide 𝑟). Hence 𝑟 is the greatest common divisor of 𝑟 and 0.

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**Greatest Common Divisor (GCD)**

Lemma 4.8.2 If 𝑎 and 𝑏 are any integers not both zero, and if 𝑞 and 𝑟 are any integers such that 𝑎=𝑏𝑞+𝑟, then gcd(𝑎,𝑏)=gcd(𝒃,𝑟) Proof: [The proof is divided into two sections: (1) proof that gcd(𝑎, 𝑏) ≤ gcd(𝑏, 𝑟 ), and (2) proof that gcd(𝑏, 𝑟 ) ≤ gcd(𝑎, 𝑏). Since each gcd is less than or equal to the other, the two must be equal.] gcd(𝒂, 𝒃) ≤ gcd(𝒃, 𝒓): [We will first show that any common divisor of 𝑎 and 𝑏 is also a common divisor of 𝑏 and 𝑟.] Let 𝑎 and 𝑏 be integers, not both zero, and let 𝑐 be a common divisor of 𝑎 and 𝑏. Then 𝑐 | 𝑎 and 𝑐 | 𝑏, and so, by definition of divisibility, 𝑎= 𝑛𝑐 and 𝑏 = 𝑚𝑐, for some integers 𝑛 and 𝑚. Now substitute into the equation 𝑎 = 𝑏𝑞 + 𝑟 to obtain 𝑛𝑐 = (𝑚𝑐)𝑞 + 𝑟.

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**Greatest Common Divisor (GCD)**

Lemma 4.8.2 If 𝑎 and 𝑏 are any integers not both zero, and if 𝑞 and 𝑟 are any integers such that 𝑎=𝑏𝑞+𝑟, then gcd(𝑎,𝑏)=gcd(𝒃,𝑟) Proof (cont’): gcd(𝒂, 𝒃) ≤ gcd(𝒃, 𝒓): [We will first show that any common divisor of 𝑎 and 𝑏 is also a common divisor of 𝑏 and 𝑟.] 𝑛𝑐 = (𝑚𝑐)𝑞 + 𝑟. Then solve for 𝑟 : 𝑟 = 𝑛𝑐 − (𝑚𝑐)𝑞 = (𝑛 − 𝑚𝑞)𝑐. But 𝑛 − 𝑚𝑞 is an integer, and so, by definition of divisibility, 𝑐 | 𝑟 . Because we already know that 𝑐 | 𝑏, we can conclude that 𝑐 is a common divisor of 𝑏 and 𝑟 [as was to be shown].

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**Greatest Common Divisor (GCD)**

Lemma 4.8.2 If 𝑎 and 𝑏 are any integers not both zero, and if 𝑞 and 𝑟 are any integers such that 𝑎=𝑏𝑞+𝑟, then gcd(𝑎,𝑏)=gcd(𝒃,𝑟) Proof (cont’): gcd(𝒂, 𝒃) ≤ gcd(𝒃, 𝒓): [Next we show that gcd(𝑎, 𝑏) ≤ gcd(𝑏, 𝑟).] By part (a), every common divisor of 𝑎 and 𝑏 is a common divisor of 𝑏 and 𝑟 . It follows that the greatest common divisor of 𝑎 and 𝑏 is defined because 𝑎 and 𝑏 are not both zero, and it is a common divisor of 𝑏 and 𝑟 . But then gcd(𝑎, 𝑏) (being one of the common divisors of 𝑏 and 𝑟) is less than or equal to the greatest common divisor of 𝑏 and 𝑟 : gcd(𝑎, 𝑏) ≤ gcd(𝑏, 𝑟 ). gcd(𝒃, 𝒓) ≤ gcd(𝒂, 𝒃): The second part of the proof is very similar to the first part. It is left as an exercise.

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**The Euclidean Algorithm**

Problem: Given two integer A and B with 𝐴>𝐵≥0, find gcd(𝐴,𝐵) Idea: The Euclidean Algorithm uses the division algorithm repeatedly. If B=0, by Lemma we know gcd(𝐴,𝐵) = 𝐴. If B>0, division algorithm can be used to calculate a quotient 𝑞 and a remainder 𝑟: 𝐴=𝐵𝑞+𝑟 where 0≤𝑟<𝐵 By Lemma 4.8.2, we have gcd(𝐴,𝐵)=gcd(𝐵,𝑟), where 𝐵 and 𝑟 are smaller numbers than 𝐴 and 𝐵. gcd(𝐴, 𝐵) = gcd(𝐵, 𝑟) = ⋯ = gcd(𝑥, 0) = 𝑥 𝑟=𝐴 𝑚𝑜𝑑 𝐵

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**The Euclidean Algorithm - Exercise**

Use the Euclidean algorithm to find gcd(330, 156).

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**The Euclidean Algorithm - Exercise**

Use the Euclidean algorithm to find gcd(330, 156). Solution: gcd(330,156) = gcd(156, 18) 330 mod 156 = 18 = gcd(18, 12) 156 mod 18 = 12 = gcd(12, 6) 18 mod 12 = 6 = gcd(6, 0) 12 mod 6 = 0 = 6

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**An alternative to Euclidean Algorithm**

If 𝑎≥ 𝑏> 0, then gcd(𝑎,𝑏)=gcd(𝑏, 𝑎−𝑏)

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**An alternative to Euclidean Algorithm**

If 𝑎≥ 𝑏> 0, then gcd(𝑎,𝑏)=gcd(𝑏, 𝑎−𝑏) Hint: Part 1: proof gcd(𝑎,𝑏)≤gcd(𝑏, 𝑎−𝑏) every common divisor of a and b is a common divisor of b and a-b Part 2: proof gcd(𝑎,𝑏)≥gcd(𝑏, 𝑎−𝑏) every common divisor of b and a-b is a common divisor of a and b.

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**Homework #5 Problems Converting decimal to rational numbers.**

4.2.2: 4.2.7: …

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Homework #5 Problems 1. Converting decimal to rational numbers : Solution: = ∗10000= : … Solution: let X = x = …. 10x = … x – 10x = X = / 99990

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Exercise Prove that for any nonnegative integer 𝑛, if the sum of the digits of 𝑛 is divisible by 9, then 𝑛 is divisible by 9. Hint: by the definition of decimal representation 𝑛= 𝑑 𝑘 10 𝑘 + 𝑑 𝑘−1 10 𝑘−1 +⋯+ 𝑑 𝑑 0 where 𝑘 is nonnegative integer and all the 𝑑 𝑖 are integers from 0 to 9 inclusive. = 𝑑 𝑘 ( 9999⋯999 𝑘 9 ′ 𝑠 +1)+ 𝑑 𝑘−1 ( 9999⋯999 𝑘−1 9 ′ 𝑠 +1)+⋯+ 𝑑 1 (9+1)+ 𝑑 0 =9 𝑑 𝑘 11⋯11 𝑘 1 ′ 𝑠 + 𝑑 𝑘−1 11⋯11 𝑘−1 1 ′ 𝑠 +⋯+ 𝑑 𝑑 𝑘 + 𝑑 𝑘−1 +⋯+𝑑 1 + 𝑑 0 = an integer divisible by 9 + the sum of the digits of 𝑛

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Homework #5 Problems Theorem: The sum of any two even integers equals 4k for some integer k. “Proof: Suppose m and n are any two even integers. By definition of even, m = 2k for some integer k and n = 2k for some integer k. By substitution, m + n = 2k + 2k = 4k. This is what was to be shown.” What’s the mistakes in this proof?

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Properties of the gcd Theorem: For any two integers a,b there exist integers x,y such that xa + yb = gcd(a,b). A proof will not be given at this point.

Properties of the gcd Theorem: For any two integers a,b there exist integers x,y such that xa + yb = gcd(a,b). A proof will not be given at this point.

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