Division and divisibility Euclidean algorithm Math 2603 - Lecture 7 Section 4.1 to 4.3 Division and prime numbers Bo Lin September 10th, 2019 Bo Lin Math 2603 - Lecture 7 Section 4.1 to 4.3 Division and prime numbers Division and divisibility For integers a; b; c, we have (commutativity) a + b = b + a; a ∗ b = b ∗ a. (associativity) a + (b + c) = (a + b) + c; a ∗ (b ∗ c) = (a ∗ b) ∗ c. (distributive law) a ∗ (b + c) = a ∗ b + a ∗ c. (identities) a + 0 = a; a ∗ 1 = a. (additive inverse) a + (−a) = 0. 1 (multiplicative inverse)a ∗ a = 1 for all a 6= 0. Remark In fact, there are other arithmetic operations. Division and divisibility Euclidean algorithm Arithmetic operations We have been familiar with addition and multiplication among real numbers. And they have a lot of good properties. Bo Lin Math 2603 - Lecture 7 Section 4.1 to 4.3 Division and prime numbers Remark In fact, there are other arithmetic operations. Division and divisibility Euclidean algorithm Arithmetic operations We have been familiar with addition and multiplication among real numbers. And they have a lot of good properties. For integers a; b; c, we have (commutativity) a + b = b + a; a ∗ b = b ∗ a. (associativity) a + (b + c) = (a + b) + c; a ∗ (b ∗ c) = (a ∗ b) ∗ c. (distributive law) a ∗ (b + c) = a ∗ b + a ∗ c. (identities) a + 0 = a; a ∗ 1 = a. (additive inverse) a + (−a) = 0. 1 (multiplicative inverse)a ∗ a = 1 for all a 6= 0. Bo Lin Math 2603 - Lecture 7 Section 4.1 to 4.3 Division and prime numbers Division and divisibility Euclidean algorithm Arithmetic operations We have been familiar with addition and multiplication among real numbers. And they have a lot of good properties. For integers a; b; c, we have (commutativity) a + b = b + a; a ∗ b = b ∗ a. (associativity) a + (b + c) = (a + b) + c; a ∗ (b ∗ c) = (a ∗ b) ∗ c. (distributive law) a ∗ (b + c) = a ∗ b + a ∗ c. (identities) a + 0 = a; a ∗ 1 = a. (additive inverse) a + (−a) = 0. 1 (multiplicative inverse)a ∗ a = 1 for all a 6= 0. Remark In fact, there are other arithmetic operations. Bo Lin Math 2603 - Lecture 7 Section 4.1 to 4.3 Division and prime numbers Definition For real numbers a; b with b 6= 0, the division is defined in term of addition that a=b = c; where c is the unique real number such that c · b = a. Today, we focus on the case when both a; b are integers. Note that c may not be an integer. But if a; b 2 Z, by definition c 2 Q. Division and divisibility Euclidean algorithm Subtraction and division Definition For real numbers a; b, the subtraction is defined in terms of addition that a − b = a + (−b): Bo Lin Math 2603 - Lecture 7 Section 4.1 to 4.3 Division and prime numbers Today, we focus on the case when both a; b are integers. Note that c may not be an integer. But if a; b 2 Z, by definition c 2 Q. Division and divisibility Euclidean algorithm Subtraction and division Definition For real numbers a; b, the subtraction is defined in terms of addition that a − b = a + (−b): Definition For real numbers a; b with b 6= 0, the division is defined in term of addition that a=b = c; where c is the unique real number such that c · b = a. Bo Lin Math 2603 - Lecture 7 Section 4.1 to 4.3 Division and prime numbers Division and divisibility Euclidean algorithm Subtraction and division Definition For real numbers a; b, the subtraction is defined in terms of addition that a − b = a + (−b): Definition For real numbers a; b with b 6= 0, the division is defined in term of addition that a=b = c; where c is the unique real number such that c · b = a. Today, we focus on the case when both a; b are integers. Note that c may not be an integer. But if a; b 2 Z, by definition c 2 Q. Bo Lin Math 2603 - Lecture 7 Section 4.1 to 4.3 Division and prime numbers Division and divisibility Euclidean algorithm Divisibility Definition If a and b are integers and b 6= 0 then a is divisible by b if and only if a equals b times some integer. Instead of a is divisible by b, we can also say that a is a multiple of b; b is a factor of a; b is a divisor of a; b divides a. The notation bja is read b divides a. Symbolically, if a and b are integers and b 6= 0 bja , 9k 2 Z such that a = k · b: Bo Lin Math 2603 - Lecture 7 Section 4.1 to 4.3 Division and prime numbers Solution (a) Since 21 = 3 · 7, yes. (b) Since 22=4 = 5:5 2= Z, no. (c) Since 28 = (−7) · (−4), yes. Division and divisibility Euclidean algorithm Examples: checking divisibility Example (a) Is 21 divisible by 3? (b) Does 4 divide 22? (c) Is 28 a multiple of −7? Bo Lin Math 2603 - Lecture 7 Section 4.1 to 4.3 Division and prime numbers (b) Since 22=4 = 5:5 2= Z, no. (c) Since 28 = (−7) · (−4), yes. Division and divisibility Euclidean algorithm Examples: checking divisibility Example (a) Is 21 divisible by 3? (b) Does 4 divide 22? (c) Is 28 a multiple of −7? Solution (a) Since 21 = 3 · 7, yes. Bo Lin Math 2603 - Lecture 7 Section 4.1 to 4.3 Division and prime numbers (c) Since 28 = (−7) · (−4), yes. Division and divisibility Euclidean algorithm Examples: checking divisibility Example (a) Is 21 divisible by 3? (b) Does 4 divide 22? (c) Is 28 a multiple of −7? Solution (a) Since 21 = 3 · 7, yes. (b) Since 22=4 = 5:5 2= Z, no. Bo Lin Math 2603 - Lecture 7 Section 4.1 to 4.3 Division and prime numbers Division and divisibility Euclidean algorithm Examples: checking divisibility Example (a) Is 21 divisible by 3? (b) Does 4 divide 22? (c) Is 28 a multiple of −7? Solution (a) Since 21 = 3 · 7, yes. (b) Since 22=4 = 5:5 2= Z, no. (c) Since 28 = (−7) · (−4), yes. Bo Lin Math 2603 - Lecture 7 Section 4.1 to 4.3 Division and prime numbers Remark This principle is equivalent to the principle of mathematical induction. In other words, either one could imply the other. Division and divisibility Euclidean algorithm The Well-ordering Principle Axiom (The Well-ordering Principle) Let S be a nonempty set of integers all of which are greater than some fixed integer C. Then S has a least element. In particular, any nonempty subset of N has a least element. Bo Lin Math 2603 - Lecture 7 Section 4.1 to 4.3 Division and prime numbers Division and divisibility Euclidean algorithm The Well-ordering Principle Axiom (The Well-ordering Principle) Let S be a nonempty set of integers all of which are greater than some fixed integer C. Then S has a least element. In particular, any nonempty subset of N has a least element. Remark This principle is equivalent to the principle of mathematical induction. In other words, either one could imply the other. Bo Lin Math 2603 - Lecture 7 Section 4.1 to 4.3 Division and prime numbers Example Suppose 3 students are devouring a large pizza together. There are 8 slices in total. How could they share the pizza as equal as possible without dividing each slice? Remark First, each student would get 2 slices. There are still 8 − 3 · 2 = 2 slices remaining. Since 2 < 3, they cannot further divide them. Remark This is exactly how division between integers works. Division and divisibility Euclidean algorithm What happens when b - a Suppose a; b are integers such that b - a, can we still do a=b? Bo Lin Math 2603 - Lecture 7 Section 4.1 to 4.3 Division and prime numbers Remark First, each student would get 2 slices. There are still 8 − 3 · 2 = 2 slices remaining. Since 2 < 3, they cannot further divide them. Remark This is exactly how division between integers works. Division and divisibility Euclidean algorithm What happens when b - a Suppose a; b are integers such that b - a, can we still do a=b? Example Suppose 3 students are devouring a large pizza together. There are 8 slices in total. How could they share the pizza as equal as possible without dividing each slice? Bo Lin Math 2603 - Lecture 7 Section 4.1 to 4.3 Division and prime numbers Remark This is exactly how division between integers works. Division and divisibility Euclidean algorithm What happens when b - a Suppose a; b are integers such that b - a, can we still do a=b? Example Suppose 3 students are devouring a large pizza together. There are 8 slices in total. How could they share the pizza as equal as possible without dividing each slice? Remark First, each student would get 2 slices. There are still 8 − 3 · 2 = 2 slices remaining. Since 2 < 3, they cannot further divide them. Bo Lin Math 2603 - Lecture 7 Section 4.1 to 4.3 Division and prime numbers Division and divisibility Euclidean algorithm What happens when b - a Suppose a; b are integers such that b - a, can we still do a=b? Example Suppose 3 students are devouring a large pizza together. There are 8 slices in total. How could they share the pizza as equal as possible without dividing each slice? Remark First, each student would get 2 slices.