Algebraic Systems, Fall 2013, September 1, 2013 Edition. Todd Cochrane


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1 Algebraic Systems, Fall 2013, September 1, 2013 Edition Todd Cochrane
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3 Contents Notation 5 Chapter 0. Axioms for the set of Integers Z. 7 Chapter 1. Algebraic Properties of the Integers Background Deducing the Additional Properties of Z from the Axioms Discreteness Axioms for Z Proof by Induction Basic Divisibility Properties Euclidean Algorithm Linear Combinations and Linear Equations Solving Linear Equations in integers Unique Factorization of Integers Further properties of primes 23 Chapter 2. Modular Arithmetic and the Modular Ring Z m A few applications of congruences Multiplicative inverses (mod m) The modular ring Z m Group of units U m and the Euler phifunction Euler s Theorem and Fermat s Little Theorem Public Key Cryptography. 31 Chapter 3. Rings, Integral Domains and Fields Basic properties of Rings Subrings of Z and Z m Polynomial Rings Zero divisors and Integral Domains Units and Fields Matrix Rings Complex Numbers nth powers and nth roots of complex numbers Subfields of the Real Numbers and Complex Numbers 45 Chapter 4. Factoring Polynomials Unique Factorization of Polynomials Factoring quadratic and cubic polynomials Useful Factoring Formulas Factoring Polynomials over C Factoring Polynomials over R 55 3
4 4 CONTENTS 4.6. Factoring Polynomials over Q Summary of Irreducible Polynomials over C, R, Q and Z p Cardano s Solution of the Cubic Equation Solution of the Quartic Equation and Higher Degree Equations. 58 Chapter 5. Group Theory Subgroups of Groups Generators and Orders of Elements Cyclic Groups The Klein4 group Direct Product of Groups Lagrange s Theorem Permutation Groups Cycle Notation Groups of Symmetries Dihedral Group D n Isomorphism. 68
5 Notation N = {1, 2, 3, 4, 5,... } = Natural numbers Z = {0, ±1, ±, 2, ±3,... } = Integers E = {0, ±2, ±4, ±6,... } = Even integers O = {±1, ±3, ±5,... } = Odd integers Q = {a/b : a, b Z, b 0} = Rational numbers R = Real numbers C = Complex numbers Z m = Ring of integers mod m [a] m = {a + mx : x Z} = Residue class of a mod m U m = Multiplicative group of units mod m a 1 (mod m) = multiplicative inverse of a (mod m) φ(m) = Euler phifunction (a, b) = gcd(a, b) = greatest common divisor of a and b [a, b] = lcm[a, b] = least common multiple of a and b a b = a divides b M 2,2 (R) = Ring of 2 2 matrices over a given ring R R[x] = Ring of polynomials over R S = order or cardinality of a set S S n = nth symmetric group intersection union empty set subset there exists! there exists a unique for all implies equivalent to iff if and only if element of congruent to 5
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7 CHAPTER 0 Axioms for the set of Integers Z. We shall assume the following properties as axioms for the set of integers. 1] Addition Properties. There is a binary operation + on Z, called addition, satisfying a) Addition is well defined, that is, given any two integers a, b, a+b is a uniquely defined integer. b) Substitution Law for addition: If a = b and c = d then a + c = b + d. c) The set of integers is closed under addition. For any a, b Z, a + b Z. d) Addition is commutative. For any a, b Z, a + b = b + a. e) Addition is associative. For any a, b, c Z, (a + b) + c = a + (b + c). f) There is a zero element 0 Z (also called the additive identity), satisfying 0 + a = a = a + 0 for any a Z. g) For any a Z, there exists an additive inverse a Z satisfying a + ( a) = 0 = ( a) + a. Properties a),b), and c) above are implicit in the definition of a binary operation. Definition: Subtraction in Z is defined by a b = a + ( b) for a, b Z. 2] Multiplication Properties. There is an operation (or ) on Z called multiplication, satisfying, a) Multiplication is well defined, that is, given any two integers a, b, a b is a uniquely defined integer. b) Substitution Law for multiplication: If a = b and c = d then ac = bd. c) Z is closed under multiplication. For any a, b Z, a b Z. d) Multiplication is commutative. For any a, b Z, ab = ba. e) Multiplication is associative. For any a, b, c Z, (ab)c = a(bc). f) There is an identity element 1 Z satisfying 1 a = a = a 1 for any a Z. 3] Distributive law. This is the one property that combines both addition and multiplication. For any a, b, c Z, a(b + c) = ab + ac. One can deduce (from the given axioms) the additional distributive laws, (a + b)c = ac + bc, a(b c) = ab ac and (a b)c = ac bc. 4] Trichotomy Principle. The set of integers can be partitioned into three disjoint sets, Z = N {0} N, where N = {1, 2, 3,... } = Natural Numbers = Positive Integers, N = { 1, 2, 3,... } = Negative Integers. One then defines the inequalities > and < by saying a > b if a b N and a < b if a b N. Thus we get the Law of Trichotomy which states that for any two integers a, b exactly one of the following holds: a < b, a = b or a > b, (that is a b N, a b = 0 or a b N.) 7
8 8 0. AXIOMS FOR THE SET OF INTEGERS Z. 5] Positivity Axiom. The sum of two positive integers is positive. The product of two positive integers is positive. 6] Discreteness Axioms. a) Well Ordering Property of N. Any nonempty subset of N has a smallest element. b) Principle of Induction. Let S be a subset of N such that (i) 1 S and (ii) n S n + 1 S. Then S = N. Additional Properties of Z. The properties below can all be deduced from the axioms above. You may assume them in your homework unless specifically asked to prove the property. See Chapter 1, Section 1.2 for proofs. 1] SubtractionEquality principle. x = y if and only if x y = 0. 2] Cancelation law for addition: If a + x = a + y then x = y. 3] Additive inverses are unique, that is, if a, b, c are integers such that a + b = 0 and a + c = 0 then b = c. 4] Zero multiplication property: a 0 = 0 for any a Z. 5] Properties of negatives: ( a)b = (ab) = a( b), ( a)( b) = ab, ( 1)a = a. 6] Basic consequence of Trichotomy: If a > 0 then a < 0 and if a < 0 then a > 0. 7] Products of Positives and Negatives: If a > 0 and b < 0 then ab < 0. If a < 0 and b < 0, then ab > 0. 8] Zero divisor property, or integral domain property: If ab = 0 then a = 0 or b = 0. 9] Cancelation law for multiplication: If ax = ay and a 0 then x = y. 10] General AssociativeCommutative Law: a) Addition: When adding a collection of n integers a 1 + a a n, the numbers may be grouped in any way and added in any order. In particular, the sum a 1 +a 2 + +a n is well defined, that is, no parentheses are necessary to specify the order of operations. b) Multiplication: When multiplying a collection of n integers a 1 a 2 a n, the numbers may be grouped in any way and multiplied in any order. In particular, the product a 1 a 2 a n is well defined, that is, no parentheses are necessary to specify the order of operations. 11] FOIL Law: For any integers a, b, c, d, (a + b)(c + d) = ac + ad + bc + bd. 12] Binomial Expansion: For any integers a, b and positive integer n we have (a + b) n = n ( n ) k=0 k a k b n k = a n + ( ) n 1 a n 1 b + ( n 2) a n 2 b b n. In particular, (a + b) 2 = a 2 + 2ab + b 2 (a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3.
9 CHAPTER 1 Algebraic Properties of the Integers 1.1. Background Definition A statement is a sentence that can be assigned a truth value. (In general there is a subject, verb and object in the statement). Example The following are statements, that is, given x R, we can definitively assert whether A, B or C is true or false: A : x 2 = 4. B : x = 2. C : x = ±2. Note that these statement really are complete sentences. subject is x 2, the verb = and the object is 4. For example in A the If A and B are statements, A B means A implies B, that is, if A is true then B is true. A B means A is equivalent to B, that is, A is true iff B is true. Example Which of the following are true for the statements in the preceding example? A C, A B, B A, A C. (T,F,T,T). Note the truth value actually depends on an implicit assumption such as x is an integer or x is a real number. If our implicit assumption is that x is a natural number, then A B is a true statement. For some algebraic systems that we shall see this semester A and C are not equivalent! Note The symbols and are used between statements. The symbol = is used between objects (numbers, functions, sets, etc. ). Be careful in making this distinction whenever you write a proof. Definition Let A, B be given sets. A function f : A B (pronounced, a function f from A to B), is a rule that assigns to each element x A a unique element f(x) B. The set A is called the domain of f and the set B, the codomain of f. The range of f, denoted f(a), is the set of all output values, The range is a subset of the codomain. f(a) := {f(x) : x A}. Definition The cartesian product of two sets A, B, denoted A B, is the set of all ordered pairs (x, y) with x A, y B. That is, A B = {(x, y) : x A, y B}. Example Z Z is the set of all ordered pairs of integers, Z Z = {(x, y) : x, y Z}. 9
10 10 1. ALGEBRAIC PROPERTIES OF THE INTEGERS Definition ) A binary operation on Z is a function : Z Z Z, that assigns to each ordered pair (a, b) of integers a unique integer denoted a b. 2) It is called commutative if a b = b a for all a, b Z. 3) It is called associative if a (b c) = (a b) c for all a, b, c Z. 4) An element e Z is called an identity element with respect to if a e = a and e a = a for all integers a. Example Ordinary addition and multiplication are binary ops on Z; so is subtraction. Division fails (why?). Addition and Multiplication are commutative and associative, and both have identities (what are they?). Definition A subset S of Z is said to be closed under (or with respect to ) if for any two a, b S we have a b S. Example Let a b = 2a + b. i) Is this a binary operation on Z? Yes, given any two integers a, b the output 2a + b is a uniquely defined integer. ii) Is this operation commutative? Note that a b = 2a + b, but b a = 2b + a. Thus a b b a in general (eg. 1 2 = 3 but 2 1 = 5.) iii) Is the operation associative? a (b c) = a (2b + c) = 2a + (2b + c) = 2a + 2b + c, whereas, (a b) c = (2a + b) c = 2(2a + b) + c = 4a + 2b + c. Since 2a + 2b + c 4a + 2b + c for a 0 we see that associativity fails. iv) Is there an identity element? Suppose that e is an identity. Then e a = a and a e = a for all a Z. Thus 2e + a = a and 2a + e = a, that is, e = 0 and e = a for all a Z. The latter condition clearly fails (e cannot equal a for all integers a.) Therefore, there is no identity. v) Is the set of odd integers O closed under? Lets check. Let a, b be odd integers. Then a b = 2a + b = even + odd = odd. Thus O is closed. Example Let a b = ab. Is this a binary op on Z? No, for example, 1 2 = 2 which is not an integer. (To be a binary operation on Z, the output has to be an integer for all possible integer inputs. If this fails for one example, then the operation fails to be a binary operation. Example The following are binary operations on Z: a b = 2b, a b = 3, a b = gcd(a 2 +1, b 2 +1), (where gcd is the greatest common divisor.) The following are not binary operations on Z: a b = b 2 /a, a b = ±a, a b = a b Deducing the Additional Properties of Z from the Axioms In this section we will deduce the Additional Properties of Z listed in Chapter 0 from the axioms. We will provide examples of two styles of proofs. The first is twocolumn style, where the right column provides the justification for each step. The second is text style, where the proof is written in paragraph form with complete sentences following all the rules of grammar. In formal mathematical writing one always uses text style, but for this class the twocolumn style is occasionally acceptable SubtractionEquality principle. For any integers x, y, x y = 0 if and only if x = y.
11 1.2. DEDUCING THE ADDITIONAL PROPERTIES OF Z FROM THE AXIOMS 11 Proof. x y = 0, (x y) + y = 0 + y, (x + ( y)) + y = 0 + y, x + ( y + y) = 0 + y, x + 0 = 0 + y, x = y, assumption addition is well defined definition of subtraction associative law additive inverse property 0 is additive identity Note that because the statement was an if and only if statement we needed leftright arrows at each step Cancelation Law for Addition. : Let a, x, y be integers such that a + x = a + y. Then x = y. Proof. a + x = a + y, assumption a + (a + x) = a + (a + y), addition is well defined ( a + a) + x = ( a + a) + y, associative law 0 + x = 0 + y, additive inverse property x = y, 0 is additive identity Note i) The following is also a version of the cancelation law: If x + a = y + a then x = y. ii) Look at the axioms required to prove the cancelation law. Any algebraic system satisfying those same axioms will also satisfy the cancelation law. Rings and Additive Groups are both examples of such systems that we will visit this semester Every integer has a unique additive inverse. Proof. (We ll do this one in text form.) By one of the axioms of Z, we know that every integer has an additive inverse, so our task here is to show that it is unique. Let a be a given integer. Suppose that b, c are additive inverses of a. Then a + b = 0 and a + c = 0. By the transitive law for equality, a + b = a + c. Thus by the cancelation law for addition (which we just proved), b = c Zero Multiplication Property. For any integer n, n 0 = 0. Proof. The formal proof is homework but we ll give you a hint. Since 0 is linked with additive properties of Z and this theorem is a multiplicative statement, you will need to make use of the one axiom linking addition and multiplication (what is it?) Now start by writing 0 = (what property have I just used?) Then use substitution to say n 0 = n (0 + 0), etc.
12 12 1. ALGEBRAIC PROPERTIES OF THE INTEGERS Properties of Negatives. For any integers a, b we have i) ( a) = a. ii) ( 1)a = a. iii) ( a)b = (ab) = a( b). iv) ( a)( b) = ab. Proof. i) Since a + ( a) = 0 = ( a) + a by the definition of additive inverse, we see that a is the additive inverse of a, that is a = ( a). ii) For this part our goal is to show that ( 1)a is the additive inverse of a, that is, ( 1)a + a = 0. Now, ( 1)a + a = ( 1)a + 1(a), iii) We have 1 is the multiplicative identity = ( 1 + 1)a, distributive law = 0a, property of additive inverses = 0, by zero mult property ( a)b = (( 1)a)b, by part (ii) = ( 1)(ab), by associativity = (ab), by part (ii) The second equality can be proven in the same manner. iv) We have ( a)( b) = (a( b)), by part (iii) = ( (ab)), by part (iii) = ab, by part (i) Basic consequence of Trichotomy. Let a Z. If a > 0 then a < 0, and if a < 0 then a > 0. Proof. Suppose that a > 0 that is, a N. Then a N and so by definition a < 0. Next, suppose that a < 0, that is, a N. Then a = c for some c N. Thus, by a property of negatives, a = ( c) = c N, and so a > Products of Positives and Negatives. i) If a > 0 and b < 0 then ab < 0. ii) If a < 0 and b < 0, then ab > 0. Proof. i) Suppose that a < 0 and b > 0. Then a = c for some c > 0, by definition of <. Thus ab = ( c)b = (cb) by a property of negatives. Now, by the Positivity Axiom, cb > 0, and thus by the preceding property, (cb) < 0, that is, ab < 0. ii) Suppose that a < 0 and b < 0. Then a = c, b = d for some positive integers c, d. Thus ab = ( c)( d) = cd by a property of negatives. By the Positivity Axiom, cd > 0, and thus ab > 0.
13 1.2. DEDUCING THE ADDITIONAL PROPERTIES OF Z FROM THE AXIOMS Zero divisor property or Integral domain property of Z. If a, b are integers with ab = 0, then a = 0 or b = 0. Proof. We ll do a proof by contradiction. Suppose that ab = 0 but a 0 and b 0. Then by trichotomy either a is positive or a is negative, and the same for b. If a, b are both positive then by the Positivity Axiom ab is positive, a contradiction. If a is positive and b is negative then ab is negative by the preceding property, a contradiction. Finally if both a and b are negative, then ab is positive by the preceding property, a contradiction. Thus, in all cases we are led to a contradiction. Therefore a = 0 or b = Cancelation Law for Multiplication. If a, x, y are integers with ax = ay and a 0, then x = y. Proof. Since we have only introduced integers at this point, we wish to prove this without using fractions. Thus we cannot simply divide both sides by a or multiply both sides by 1/a. Instead, we will make use of the subtraction equality principle and the integral domain property of Z. The details are left for your homework. Note Be careful in your use of the symbols = and when writing a proof. Note, the equal symbol is used between objects (equal numbers, equal sets, equal functions, etc.), whereas the symbols and are used between statements (remember a statement is a sentence that can be assigned a truth value, true or false.) General AssociativeCommutative Law. a) Addition: When adding a collection of n integers a 1 + a a n, the numbers may be grouped in any way and added in any order. In particular, the sum a 1 +a 2 + +a n is well defined, that is, no parentheses are necessary to specify the order of operations. b) Multiplication: When multiplying a collection of n integers a 1 a 2 a n, the numbers may be grouped in any way and multiplied in any order. In particular, the product a 1 a 2 a n is well defined, that is, no parentheses are necessary to specify the order of operations. Note We will not attempt to prove this law here, as it requires a rather sophisticated use of induction. Instead, lets just gain some appreciation of what it is saying, since we will be making extensive use of it throughout the semester. What does a + b + c + d mean? Remember, addition is a binary operation, that is you can only add two integers at a time. There are many possible definitions, ((a+b) +c) +d, (a+(b+c)) +d, (a+b) +(c+d), a+((b+c) +d), a+(b+(c+d)) and so on. The general associative law tells us that all of these expressions are equal, and thus there is no need to include the parentheses at all. For instance, we can see that the first two expressions in the list are plainly equal by one application of the associative law, (a + b) + c = a + (b + c). If we throw in the word commutative, then the general associativecommutative law tells us that we can also rearrange the order. Thus for example (d + b) + (a + c) would also equal a + b + c + d. A similar discussion holds for multiplication. We can really appreciate this law when working with rational numbers. For example try calculating the following in your head: What is the easiest way to do it?
14 14 1. ALGEBRAIC PROPERTIES OF THE INTEGERS The FOIL law. For any integers a, b, c, d, (a + b)(c + d) = ac + ad + bc + bd. Proof. We have (a + b)(c + d) = (a + b)c + (a + b)d, distributive law = (ac + bc) + (ad + bd), distributive law = ac + ad + bc + bd, general associativecommutative law Binomial Square Formula. For any positive integer n and integers a, b we have (a + b) 2 = a 2 + 2ab + b 2. We have (a + b) 2 = (a + b)(a + b), definition of square = a 2 + ba + ab + b 2, FOIL law = a 2 + ab + ab + b 2, commutative law for mult = a 2 + (ab + ab) + b 2, general associative law = a 2 + 2ab + b 2, definition of 2 times a number. We shall prove the general binomial expansion formula using induction in Section Discreteness Axioms for Z Let us return now to the two discreteness axioms for Z. These are the axioms that distinguish the integers from sets such as Q and R, which also satisfy all of the algebraic axioms (associative law, commutative law, distributive law, etc. ) These axioms imply that the integers are discrete objects. For Q and R we can say that between any two elements of the set there are infinitely many other elements of the set. Thus there is no gap between one rational or real number and the next one. For integers this is false. For instance, between 0 and 1 there are no other integers. More generally, for any distinct integers a, b we can say a b 1. a) Well Ordering Property of N. Any nonempty subset of N has a smallest element. Note that this property does not hold for the set of positive rational numbers Q + or positive real numbers R +. Consider for example the interval of real numbers (0, 1). This set has no smallest element. b) Axiom of Induction. Let S be a subset of N such that (i) 1 S and (ii) n S n + 1 S. Then S = N. Again, it is plain that this axiom fails for Q + and R +. One can prove that these two axioms are equivalent, that is the well ordering property of N implies the axiom of induction, and the axiom of induction implies the well ordering property. (See if you can prove either direction!) Here are a couple more equivalent discreteness properties that we will occasionally appeal to, but will not prove here.
15 1.4. PROOF BY INDUCTION 15 c) Maximum Element Principle. Any nonempty subset of integers bounded above contains a maximum element. d) Minimum Element Principle. Any nonempty subset of integers bounded below contains a minimum element Proof by Induction An important method of proof that we shall use in this class is a variation of the axiom of induction that we call the principle of induction. It is used for proving that a given statement is true for all natural numbers. Principle of Induction. Let P (n) be a statement involving a natural number n. Suppose that (i) P (1) is true. (Base Case.) (ii) If P (n) is true for a given n N then P (n + 1) is true. (Inductive Step.) Then P (n) is true for all n N. The assumption P (n) is true for a given n N is called the induction assumption. Note One of the common errors in proving something is to assume the statement you wish to prove is true in the middle of the proof. How would you respond to someone who objects to the Principle of Induction by saying in the induction assumption you are assuming what you wish to prove? (Note the subtle distinction. In the induction assumption, although n is arbitrary, we are only assuming P (n) is true for one value of n, not for all integers n.) Example Prove that for any positive integer n, (1.1) n 3 = n2 (n + 1) 2. 4 Proof. Proof by induction. For n = 1 we have 1 3 = , a true statement. Suppose that statement (1.1) is true for a given n. Then for n + 1 we have n 3 + (n + 1) 3 = ( n 3 ) + (n + 1) 3 = n2 (n + 1) 2 + (n + 1) 3, by induction assumption (1.1). 4 (Lets interrupt the proof with a little motivation. In your formal writeup you do not need to include these comments. Our goal is to establish the truth of (1.1) for n + 1, that is, we are hoping to get (n + 1) 2 (n + 2) 2 /4. Since this expression is in factored form, we proceed by factoring, rather than expanding.) (n + 1)2 = [n 2 + 4(n + 1)], 4 (n + 1)2 = [n 2 + 4n + 4] = 4 (n + 1)2 4 [n + 2] 2 = (n + 1)2 ((n + 1) + 1) 2. 4 Thus (1.1) holds for n + 1. At this point, there are two ways to conclude the induction proof. You can either say Thus, by the Principle of Induction, the
16 16 1. ALGEBRAIC PROPERTIES OF THE INTEGERS statement is true for all n N, or you can simply write QED, which stands for the Latin expression quod erat demonstrandum meaning literally what was to be demonstrated, but is more liberally taken to mean thus we have established what we wished to prove. In this example you should also try restating everything in sigma notation. The statement in this notation would read n k=1 k3 = n2 (n+1) 2 4 for any n N. Example n 3 n is a multiple of 3 for any positive integer n. Proof. Proof by induction. For n = 1 we note that = 0 = 0 3, a multiple of 3. Suppose that the statement is true for a given n, that is, n 3 n = 3k for some k Z. Then for n + 1 we have (n + 1) 3 (n + 1) = n 3 + 3n 2 + 3n + 1 n 1 = (n 3 n) + 3n 2 + 3n = 3k + 3n 2 + 3n, by induction assumption, = 3(k + n 2 + n) = 3 integer, since the integers are closed under addition and multiplication. QED. Example n 1 is a multiple of 5 for any positive integer n. Proof. Proof by induction. For n = 1, 6 n 1 = 6 1 = 5, a multiple of 5. Suppose that the statement is true for a given n, that is, 6 n 1 = 5k for some integer k. Then for n + 1 we have, 6 n+1 1 = 6 n 6 1 = (5k + 1)6 1, by the induction hypothesis. Then, using the distributive law we see that 6 n+1 1 = 30k = 30k + 5 = 5(6k + 1), a multiple of 5, since 6k + 1 is an integer. Thus the statement is true for n + 1. QED. Example The word induction is connected to the concept of inductive reasoning, a type of reasoning where one looks at data and tries to find a pattern or rule governing the data. Try the following example. Look at the sum of the first n odd numbers for n = 1, 2, 3, 4, 5: 1=1, 1+3=4, 1+3+5=9, =16, =25. What is the pattern? Write down a conjecture for what you think (2n 1) equals in general, and then prove it by induction. Example The Fibonacci sequence {F n } = 1, 1, 2, 3, 5, 8, 13,..., is governed by the rule F n+1 = F n + F n 1 for n 2, and the initial values F 1 = F 2 = 1. Prove that (1.2) F 1 + F F 2k 1 = F 2k, for any k N. Proof. Proof by induction on k. For k = 1 we have F 1 = 1 = F 2, so the statement is true. Suppose that the statement (1.2) is true for a given k. Then for
17 1.4. PROOF BY INDUCTION 17 k + 1 we have F 1 + F F 2k 1 + F 2k+1 = (F 1 + F F 2k 1 ) + F 2k+1 = F 2k + F 2k+1, by the induction hypothesis, = F 2k+2 = F 2(k+1), by the defining property of the Fibonacci sequence. QED Property 11. Binomial Expansion Formula. For any positive integer n and integers a, b we have (1.3) n ( ) ( ) ( ) ( ) n n n n (a+b) n = a k b n k = a n + a n 1 b+ a n 2 b ab n 1 +b n. k 1 2 n 1 k=0 Proof. The proof is by induction on n. For n = 1 the statement is trivial, (a + b) 1 = a + b. Suppose the statement is true for a given n. Then for n + 1 we have n ( ) n (a + b) n+1 = (a + b)(a + b) n = (a + b) a k b n k k = n k=0 ( ) n a k+1 b n k + k l=1 n k=0 k=0 ( ) n a k b n+1 k k ( ) n 1 n ( ) ( ) n n n = a n+1 + a k+1 b n k + b n+1 + n k 0 k=0 k=1 n ( ) n n ( n = a n+1 + b n+1 + a l b n+1 l + l 1 l l=1 l=1 n (( ) ( )) n n = a n+1 + b n a l b n+1 l l 1 l l=1 l=0 ( ) n a k b n+1 k k ) a l b n+1 l n ( ) n+1 n + 1 ( ) n + 1 = a n+1 + b n+1 + a l b n+1 l = a l b n+1 l, l l Strong Form of Induction. A variation of induction that we will sometimes use is called the Strong Form of Induction given below. It has the advantage in that one is allowed to assume a lot more in the induction assumption. We will see it used when we prove the Fundamental Theorem of Arithmetic. Strong Form of Induction. Let P (n) be a statement involving a natural number n. Suppose that (i) P (1) is true. (Base Case.) (ii) If P (k) is true for all k < n, for a given n N, then P (n) is true. (Inductive Step.) Then P (n) is true for all n N.
18 18 1. ALGEBRAIC PROPERTIES OF THE INTEGERS 1.5. Basic Divisibility Properties Our goal is to prove the Fundamental Theorem of Arithmetic, which states that every positive integer can be uniquely expressed as a product of primes, but to get there we need to start with basic properties of divisibility. Definition Let a, b Z, a 0. ax = b for some integer x. We say a divides b, written a b, if Example since 12 = 3 4; 5 12 since 12/5 / Z. Distinguish 3 12 from 3/12: the first is a statement and the latter an object. Note There are many equivalent ways of expressing the statement a divides b: a is a divisor of b, a is a factor of b, b is divisible by a, b is a multiple of a, b/a is an integer. Note, the latter form assumes knowledge about the rational numbers. At this point in the semester, I want you to prove statements about the integers without making reference to the larger number system Q. Example a. What are the divisors of 6? {±1, ±2 ± 3 ± 6}. b. What are the divisors of 0? All integers (except 0). (Ruling 0 out is just a technical assumption in our definition of divisibility above (a 0). It might make sense to say 0 is a divisor of 0 since 0 = 0 0, indeed 0 = 0 b for any b Z. It is ruled out because 0/0 is an undefined quantity.) Theorem Basic divisibility properties. Let a, b, d be integers. (i) If d a and d b then d (a + b). (ii) If d a and d b then d (a b). (iii) If d a and d b then for any integers x, y, d (ax + by). Proof. (iii) Suppose that d a, d b and that x, y Z. Then a = dk and b = dl for some integers k, l. Thus, ax + by = (dk)x + (dl)y = d(kx) + d(ly) = d(kx + ly) = d(integer), since Z is closed under addition and multiplication. Thus d ax + by. Example Another way to think about the basic divisibility properties, is to use the word multiple. Property (i) says that if a and b are multiples of d then so is a + b, while (ii) says that if a and b are multiples of d then so is a b. For example, if a and b are multiples of 5 then so are a + b and a b. Another way yet of saying this is the following: If S is the set of all multiples of 5, then S is closed under addition and subtraction. Theorem Transitive law for divisibility. For any integers a, b, c, if a b and b c, then a c. Proof. Homework Definition Let a, b be integers not both 0. The greatest common divisor of a, b, denoted gcd(a, b) is the largest integer that divides both a and b. ii) Two numbers are called relatively prime if gcd(a, b) = 1. Note gcd(0,0) is undefined. Why? 2. If a, b are not both zero, gcd(a,b) exists and is unique. (Why? Let S be the set of positive common divisors. It is a finite nonempty set, so it has a maximum element by the discreteness property of Z.)
19 1.6. EUCLIDEAN ALGORITHM gcd(0, n) = n. 4. gcd(a, b) = gcd(b, a)=gcd( a, b)= gcd( a, b). Example ) gcd(16,28)=4. 2) gcd(6,16,28) = Euclidean Algorithm. The Euclidean algorithm, an efficient way of computing GCDs, is based on two theorems, the Subtraction Principle for GCDs and the Division Algorithm. Theorem Subtraction Principle for GCDs. For any a, b Z, not both zero, and any integer q, gcd(a, b) = gcd(a qb, b). Proof. Let S be the set of common divisors of a and b, and T the set of common divisors of a qb and b. We claim that S = T, and so S and T have the same maximal element, that is, gcd(a, b) = gcd(a qb, b). To show S = T we need to show S T and T S. These inclusions follow readily from the basic divisibility properties. (Fill in details.) Example Find gcd(1023, 1026). By subtraction principle this equals gcd(1023, 3). The latter equals 3 since Division of Integers with remainder. Ex = 7R3, that is, 38 = Recall, 7 is called the quotient, 3 the remainder, 38 the dividend and 5 the divisor. Ex = 4R4, that is, 24 = ( 4) Ex. 3 8 = 0R3, that is, 3 = Note the remainder is always nonnegative and strictly smaller than the divisor. Theorem Division Algorithm. Let a, b be integers with b > 0. Then there exist integers q, r such that a = qb + r with 0 r < b. Moreover q, r are unique. (q=quotient and r= remainder in dividing a by b.) Proof. Existence: Let q be the greatest integer such that qb a, so that qb a < (q + 1)b. Then set r = a qb. It is easy to see that a = qb + r and that 0 r < b. Uniqueness: If a = qb+r = q b+r with 0 r, r < b, then b q q = r r < b and so q q < 1. Thus q = q and consequently r = r. (Fill in the details!) We are now ready to describe the Euclidean Algorithm with an example. (Recall, an algorithm is a step by step procedure for carrying out some task.) Example Find d = gcd(126, 49), by the Euclidean Algorithm. To get started we calculate = 2R28 by long division, and so 126 = Then, by the subtraction principle for GCDs, gcd(126, 49) =gcd( , 49) =gcd(28, 49). We now repeat the process by calculating 49 28, etc. (1) 126 = , d = gcd(28, 49) (2) 49 = , d = gcd(28, 21) (3) 28 = , d = gcd(7, 21) (4) 21 = 3 7, d = gcd(7, 0) = 7, ST OP The process stops when you get a remainder of 0.
20 20 1. ALGEBRAIC PROPERTIES OF THE INTEGERS 1.7. Linear Combinations and Linear Equations Definition A linear combination (LC) of two integers a, b is an integer of the form ax + by where x, y Z. Claim: If d = gcd(a, b) then d can be expressed as a linear comb. of a and b, that is, the equation (1.4) ax + by = d, has a solution in integers x, y. Example gcd(20,8)=4. By trial and error, 4 = ( 2)8. gcd(21,15)=3. By trial and error, 3 = We will see two methods for solving the GCD equation (1.4). The first is the method of Back Substitution and the second, the Array Method. Back Substitution: A method of solving the equation d = ax + by (with d = gcd(a, b)) by working backwards through the steps of the Euclidean algorithm. Example Use example above for gcd(126,49) to express 7 as a LC of 126 and 49. Use the method of back substitution. Start with equation (3): 7 = By (2) we have 21 = Substituting this into previous yields 7 = 28 (49 28) = By (1) we have 28 = Substituting this into previous yields 7 = 2 ( ) 49 = , QED. Array Method. A method for solving the linear equation ax + by = c for any c Z. Here we will do it for the case where c = gcd(a, b). Example We shall redo the previous example using the array method. To begin, set up an array with the first three columns initialized as shown below. For a given choice of x and y the linear combination 126x + 49y is given in the first row. Now, perform the Euclidean Algorithm on the numbers in top row, but do the corresponding column operations on the entire array. Let C 1 be the column with top entry 126, C 2 the column with top entry 49, etc.. The first step in the Euclidean algorithm is to subtract 2 times 49 from 126, so we let the next column C 3 be given by C 3 = C 1 2C 2. Then C 4 = C 2 C 3, C 5 = C 3 C x + 49y x y Thus, 7 = Example Find gcd(83, 17) and express it as a LC of 83 and x + 17y x Thus gcd = 1 and 1 = y By applying these methods to an arbitrary pair of integers a, b, we obtain the following theorem, called the GCDLCtheorem, Greatest Common Divisor Linear Combination Theorem. Theorem GCDLC theorem. Let a, b be integers not both zero, d = gcd(a, b). Then d can be expressed as a linear combination of a and b.
21 1.9. UNIQUE FACTORIZATION OF INTEGERS 21 Proof. (Optional Reading) A constructive proof can be given by following the Euclidean Algorithm together with the method of back substitution. The notation is rather cumbersome however. We shall give here instead a nonconstructive proof. Let S = {ax + by : x, y Z}, the set of all linear combinations of a and b. This set clearly contains positive integers, so let e be the smallest positive integer in the set (e exists by well ordering). Say e = ax 0 + by 0, for some x 0, y 0 Z. We claim that e = d. Since d a and d b, we know d e, by a basic divisibility property. In particular, d e. Thus, it suffices to show that e is a common divisor of a and b, for this would imply that e d, the greatest common divisor of a and b. Lets show that e a. To do this, we shall compute a e and show that the remainder is 0. By the division algorithm, a = qe + r, for some q, r Z with 0 r < e. Thus a = q(ax 0 + by 0 ) + r, so r = a(1 qx 0 ) bqy 0 a linear combination of a and b. Since r < e we must have r = 0 by the minimality of e in S. Therefore e a. In the same manner we obtain e b. QED Corollary GCDLC corollary. Let d = gcd(a, b). (i) The set of all linear combinations of a, b is just the set of multiples of d. (ii) The gcd of a and b is the smallest positive linear combination of a and b. Proof. (i) Suppose that e is a LC of a, b, so that, e = ax+by for some x, y Z. Since d a and d b we must have d (ax + by) by basic divisibility property. Thus d e, that is e is a multiple of d. Conversely, suppose that e is a multiple of d, say e = dk for some k Z. By GCDLC theorem we know d = ax + by for some x, y Z. Thus e = kd = k(ax + by) = (kx)a + (ky)b a LC of a and b. (ii) This follows immediately from the fact that every LC of a and b is a multiple of d, and the smallest positive multiple of d is d Solving Linear Equations in integers Suppose that we wish to solve the equation ax + by = c in integers x, y. The preceding corollary tells us that this equation can be solved iff c is a multiple of d, where d = gcd(a, b), that is d c. This gives us Theorem Solvability of a Linear Equation. The linear equation ax + by = c has a solution in integers x, y iff d c where d = gcd(a, b). Example Solve the following equations or show that there is no solution. 120x 75y = 150,, 120x 75y = 11. By the array method we obtain 120(2) 75(3) = 15, the gcd of 120 and 75. Multiplying by 10 gives the solution (20, 30) to the first equation above. Since the second equation has no solution. Example A parcel costs $2 and we only have 13 cent and 17 cent stamps. How can we do it? 13x + 17y = 200. We know 200 is a LC since gcd(13,17)=1. Using the array method we obtain the solution (50,50). Then note that you can keep adding (17,13) to get other solutions. Keep adding this until you obtain a solution that makes sense (that is, both x and y are positive.) 1.9. Unique Factorization of Integers Definition Two integers a, b are called relatively prime if gcd(a, b) = 1. Lemma Euclid s Lemma. If d ab and gcd(d, a) = 1 then d b.
22 22 1. ALGEBRAIC PROPERTIES OF THE INTEGERS Note: This lemma fails if gcd(d, a) 1. For example 4 (2 2), but 4 2. Thus d ab does not imply that d a or d b. Note Applications of Euclid s Lemma. (i) Every rational number can be uniquely expressed as a fraction in reduced form. Proof. Homework. (ii) If n is not a perfect square, then n is irrational. Proof. Homework. Definition i) A positive integer p > 1 is called a prime if its only positive factors are 1 and itself. 2,3,5,7,... ii) A positive integer n > 1 is called a composite if it is not a prime, that is, n = ab for some positive integers a, b with a > 1 and b > 1. 4,6,8,9,... Note is not a prime or a composite. It is the multiplicative identity element. (Later, we will call it a unit.) Why? If 1 is a prime then we would violate unique factorization, eg 6 = 2 3 = Lemma a) Let p be a prime such that p ab. Then p a or p b. b) Let p be a prime such that p a 1 a 2... a k where a i are integers. Then p a i for some i. Proof. Use Euclid s lemma for part (a) and induction for (b). Theorem FTA: Fundamental Theorem of Arithmetic. Any positive integer n > 1 can be expressed as a product of primes, and this expression is unique up to the order of the primes. Note (i) 12 = = = 3 2 2, are all considered the same factorization. (ii) We say that a prime p has a trivial factorization as a product of primes. Proof of FTA. Existence. Proof is by strong form of induction. Let P (n) be the statement that n has a factorization as a product of primes. P (2) is true. Suppose P (k) is true for all values k smaller than a given n. Consider P (n). If n is prime we are done. Otherwise n = ab for some integers a, b with 1 < a < n, 1 < b < n. By the induction assumption, a and b can be expressed as products of primes, say a = p 1 p k, b = q 1 q l. Then ab = p 1 p k q 1 q l, a product of primes. QED Uniqueness. Suppose that n is a positive integer with two representations as a product of primes, say, (1.5) n = p 1 p k = q 1 q r for some primes p i, q j, 1 i k, 1 j r. We may assume WLOG that k r. Then p 1 q 1... q r, so by lemma, p 1 q i1 for some i 1 {1, 2,..., r}. Since p 1 and q i1 are primes, we must have p 1 = q i1. Cancelling p 1 in (1.5) yields (1.6) p 2 p 3 p k = q 1 ˆq i1 q r, where ˆq i1 indicates that this factor has been removed. We can then repeat the argument with p 2 in place of p 1. After repeating this process k times we have that (1.7) p 1 = q i1, p 2 = q i2,..., p k = q ik for some distinct integers i 1, i 2,..., i k {1, 2,..., r}. Moreover, after cancelling each of the p i from (1.5) we are left with 1 on the LHS. If r > k then (1.5) would
23 1.10. FURTHER PROPERTIES OF PRIMES 23 say that 1 is a product of primes, a contradiction. Therefore r = k, and so by (1.7), the primes p i are just a permutation of the primes q i Further properties of primes Theorem There exist infinitely many primes. Proof. (Euclid) Proof by contradiction. Suppose that there are finitely many primes, say {p 1, p 2,..., p k }. Let N = p 1 p 2 p k +1. By FTA, N has a prime factor p i, for some i k. Thus, p i N and p i (p 1 p 2 p k ). Therefore p i (N p 1 p k ), that is, p i 1, a contradiction. Theorem Basic primality test. Let a > 1 be a positive integer such that a is not divisible by any prime p with p a. Then a is a prime. Proof. Proof by contradiction. Suppose that n is composite, say n = ab with 1 < a < n, 1 < b < n. We claim that either a n or b n, else ab > n n = n = ab, a contradiction. Say a n. Let p be any prime divisor of a. Then p a n, and, since p a and a n we have p n. But this contradicts assumption that n has no prime divisor p n. Therefore n is a prime. The Sieve of Eratosthenes: This is the method of finding all of the primes in a given interval [a, b] by crossing out (sieving) all multiples of primes p b. Example Find all primes between 200 and 220. Start by making a list of all the integers from 200 to 220, then cross out all multiples of 2,3,5,7, 11 and 13. Since 17 2 = 289 > 220 we don t need to consider 17 or any larger prime. Also, note that we don t need to cross out multiples of composites such as 4,6,8,9,.. since they already have smaller prime factors. At the end of this process, the only values left in the array must be primes by the preceding theorem.
24
25 CHAPTER 2 Modular Arithmetic and the Modular Ring Z m Example What s the pattern? 3+5=8, 6+4=10, 7+6=1, 9+8=5, 9+2=11 Let m N. m =modulus. Definition We say that two integers a, b are congruent modulo m, written a b (mod m), if a and b differ by a multiple of m, that is m (a b). Note: a b (mod m) is equivalent to a = b + mk for some integer k. Example Let m = 12. Then 16 4 (mod 12) since 16 4 = (mod 12). In the example above we see = 17 5 (mod 12). How about 256 what is it (mod 12). 256 = , so (mod 12). Definition The least residue of a (mod m) is the smallest nonnegative integer that a is congruent to (mod m). Note: The least residue of a (mod m) is the remainder in dividing a by m. Since 0 r < m l.r. is always in {0, 1, 2, 3,..., m 1}. Example m = 5 Wrap the integers around a five hour clock. Theorem Congruence is an equivalence relation. That is (i) Reflexive, (ii) Symmetric and (iii) Transitive. Theorem Important properties of congruences. The substitution laws. Suppose a b (mod m), and c d (mod m). Then (i) a ± c b ± d (mod m). (ii) a c b d (mod m). (iii) a n b n (mod m) for any positive integer n. Proof. (i) a b (mod m) m (a b). c d (mod m) m (d c). Thus, by a basic divisibility property, m [(a b) + (d c)], and so, by the associative and commutative laws, m [(a + d) (b + c)], that is, a + d b + c (mod m). (ii) We ll do this one in a different style. a b (mod m) a = b + mk for some k Z. c d (mod m) c = d + ml for some l Z. Thus ac = (b + mk)(d + ml) = bd + mkd + bml + mkml = bd + m(kd + bl + kml), by the distributive, commutative and associative laws. Since kd + bl + kml Z we see that ac and bd differ by a multiple of m, that is ac bd (mod m). (iii) The proof is by induction on n. For n = 1 the statement is trivially true. Suppose the statement is true for n, and now consider n + 1. Then we have a b (mod m) and by the induction assumption a n b n (mod m). Thus by property (ii), a a n b b n (mod m), that is, a n+1 b n+1 (mod m). QED 25
26 26 2. MODULAR ARITHMETIC AND THE MODULAR RING Z m Example (mod 7) (mod 5). Note that for a chain of congruences, the modulus (mod 5) is only indicated once on the far right. Example Explore powers of 2 (mod 3), (mod 6), (mod 7), (mod 8), (mod 9), etc.. For instance working (mod 6) we have 2 1, 2 2, 2 3, = 2, 4, 2, 4, 2,..., whereas (mod 7) we get 2 1, 2 2, 2 3, = 2, 4, 1, 2, 4, 1,.... Find the length of the repeating pattern in each case. Note that the repeating pattern always has length less than the modulus. Use the pattern discovered for (mod 6) and (mod 7) to calculate (mod 6) and (mod 7). Answers: 4, 2. Note Standard trick for calculating a n (mod m) if gcd(a, m) = 1. First find a power k such that a k ±1 (mod m). We will see a theorem called Euler s theorem later on that will give us an explicit value for such a k. For now, we will just use computation as in the previous example to find such a k. Example i) Find (mod 5). First note that 47 2 (mod 5), then compute 2 1, 2 2, 2 3, = 2, 4, 3, 1, 2,... to see that (mod 5). Thus (2 4 ) (mod 5). ii) Find (mod 7). This time we note that (mod 7) and so (2 3 ) (mod 7). iii) Find (mod 17). This time we observe that (mod 17) and so (2 4 ) 25 ( 1) (mod 17) A few applications of congruences Example Day of the week. What day of the week is it 10 years from today? (Hint: Work (mod 7), and don t forget the leap years.) What time will it be 486 hours from now? (Hint: Use a 24 hour clock.) Example Divisibility tests. We will assume that every positive integer n has a unique decimal representation (2.1) n = a k 10 k + a k 1 10 k a a 0 where the a i are the digits of n, a i {0, 1, 2,..., 9}, a k 0. In standard form, n would be written n = a k a k 1... a 0, but we will avoid this notation to avoid confusion with the product of the digits. (The proof that every positive integer has such a representation can be done using the strong form of induction. ) Thus, for example 2715 = Theorem Divisibility tests for 3,9 and 11. Let n be a positive integer with decimal representation as in (2.1). (i) 3 n iff 3 (a k + + a 0 ). (ii) 9 n iff 9 (a k + + a 0 ). (iii) 11 n iff 11 (a k a k 1 + a k 2 + ( 1) k a 0 ). Proof. (ii) We ll do the test for 9, the others being similar. First we observe that by the substitution properties for congruences, since 10 1 (mod 9), we have n a k 1 k + a k 1 1 k a 0 a k + a k a 0 (mod 9). Thus n 0 (mod 9) if and only if a k + a 0 0 (mod 9), that is 9 n iff 9 (a k + + a 0 ).
27 2.3. THE MODULAR RING Z m 27 Example The UPC symbol is a 12 digit code d 1, d 2,..., d 12, where the check digit d 12 is chosen such that 3(d 1 +d 3 + +d 11 )+(d d 12 ) 0 (mod 10). This extra digit is included to prevent errors in the scanning or human input of the UPC digits. If the congruence fails after inputting the digits then you will know there is an error in the input. However, if the congruence holds, you are not guaranteed that the input is correct Multiplicative inverses (mod m) Definition An integer x is called a multiplicative inverse of a (mod m) if ax 1 (mod m). We write x a 1 (mod m) in this case. (Fraction notation 1 a is not used in modular arithmetic.) Example Find the multiplicative inverse of 3 (mod 5), 4 (mod 6), by trial and error. Which numbers have multiplicative inverses (mod 10). 1. Theorem a has a multiplicative inverse (mod m) if and only if gcd(a, m) = Example Find the multiplicative inverse of 12 it to solve the linear congruence 12x 5 (mod 17). Example Solve 3x 5 (mod 6). (mod 17) and then use Theorem Cancelation Law for modular arithmetic. Suppose that ax ay (mod m) and that gcd(a, m) = 1. Then x y (mod m). Theorem The congruence ax b (mod m) is solvable if and only if d b where d = gcd(a, m). Proof. Homework The modular ring Z m Definition The (residue class) congruence class of a (mod m), denoted [a] m is the set of all integers congruent to a (mod m). Thus [a] m = {a + km : k Z}. Example [2] 5 = {2, 7, 12,... } { 3, 8,... }. Note [7] 5, [12] 5 also represent the same class. Draw a five hour clock and observe the different residue classes at each of the five hours. Note [a] m = [b] m if and only if a b (mod m). Thus eg. [2] 5 = [12] 5. The values 2,7,12, etc. are called representatives for the class [2] 5. Definition (i) Let m be a positive integer. The ring of integers (mod m) (also called the modular ring or residue class ring (mod m)) denoted Z m, is the set of all congruence classes (mod m), Z m = {[0] m,..., [m 1] m }, together with the addition and multiplication laws defined in (ii). (ii) We define addition and multiplication on Z m as follows: For [a] m, [b] m Z m, [a] m + [b] m := [a + b] m, [a] m [b] m := [ab] m.
28 28 2. MODULAR ARITHMETIC AND THE MODULAR RING Z m Example [3] 5 + [4] 5 = [2] 5. [3] 5 [4] 5 = [2] 5. Note Addition and multiplication are well defined on Z m, that is, if [a] m = [b] m and [c] m = [d] m then [a + c] m = [b + d] m and [ac] m = [bd] m. (That is, the sum and product do not depend on the choice of representatives for the congruence classes.) Proof. We ll do multiplication. The proof for addition is similar. First, the definition of multiplication in Z m is [x] n [y] m = [xy] m, for any [x] m, [y] m Z m. To show that the product is well defined we must show that the product does not depend on the choice of representatives for the congruence classes. Now lets begin the proof. Suppose that [a] m = [a ] m and [b] m = [b ] m. Our goal is to show that [ab] m = [a b ] m. By the definition of a congruence classes, we have a a (mod m) and b b (mod m). By the substitution property of congruences this implies that ab a b (mod m), that is, [ab] m = [a b ] m. QED. Note (i) The following algebraic axioms for Z hold for Z m as well: Commutative, Associative, Distributive, zero element, additive inverses. (ii) Note one important property that Z has that Z m doesn t have in general: Integral domain property. If m is composite and xy = 0 in Z m, we cannot conclude that x = 0 or y = 0. We will return to this in the next chapter. Convention. If it is understood that we are working in Z m then the bracket notation can be dropped. Thus we can abbreviate Z m = {0, 1, 2,..., m 1}, and we can say things like in Z 6, 3 7 = 3. What is in Z 5? Answer: 2. The example of clockarithmetic that we started this chapter with is abbreviated notation in Z 12. Example Make an addition table and multiplication table for Z 4 using the abbreviated notation Group of units U m and the Euler phifunction Definition The group of units having multiplicative inverses in Z m. (mod m), U m, is the set of x Z m Note (i) We saw earlier that an element a Z m has a multiplicative inverse iff a is relatively prime to m. Thus U m is the set of elements a Z m with gcd(a, m) = 1. (ii) U m is closed under multiplication. Why? If a, b have multiplicative inverses a 1, b 1 respectively, then ab has a multiplicative inverse (ab) 1 = a 1 b 1. Example Below is the multiplication table for U 9.
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