NUMBER THEORY Prepared by Engr. JP Timola Reference: Discrete Math by Kenneth H. Rosen
NUMBER THEORY Study of the set of integers and their properties
DIVISIBILITY Division When one integer is divided by a second nonzero integer, the quotient may or may not be an integer. For example, 12/3 = 4 is an integer, whereas 11/4 = 2.75 is not.
DIVISIBILITY Definition 1 If a and b are integers with a = 0, we say that a divides b if there is an integer c such that b = ac, or equivalently, if b/a is an integer. When a divides b we say that a is a factor or divisor of b, and that b is a multiple of a. The notation a b denotes that a divides b. We write a b when a does not divide b.
EXAMPLE Determine whether 3 7 and whether 3 12.
Theorem 1 DIVISIBILITY
The Division Algorithm When an integer is divided by a positive integer, there is a quotient and a remainder
The Division Algorithm THEOREM 2 THE DIVISION ALGORITHM Let a be an integer and d a positive integer. Then there are unique integers q and r, with 0 r < d, such that a = dq + r.
The Division Algorithm Definition 2 In the equality given in the division algorithm, d is called the divisor, a is called the dividend, q is called the quotient, and r is called the remainder. This notation is used to express the quotient and remainder: q = a div d, r = a mod d.
EXAMPLE 1 What are the quotient and remainder when 101 is divided by 11?
EXAMPLE 2 What are the quotient and remainder when 11 is divided by 3?
MODULAR ARITHMETIC Modular arithmetic operates with the remainders of integers when they are divided by a fixed positive integer, called the modulus
Modular Arithmetic DEFINITION 3 If a and b are integers and m is a positive integer, then a is congruent to b modulo m if m divides a b. We use the notation a b (mod m) to indicate that a is congruent to b modulo m.we say that a b (mod m) is a congruence and that m is its modulus (plural moduli). If a and b are not congruent modulo m, we write a b (mod m).
THEOREM 3 Let a and b be integers, and let m be a positive integer. Then a b (mod m) if and only if a mod m = b mod m.
EXAMPLE Determine whether 17 is congruent to 5 modulo 6 and whether 24 and 14 are congruent modulo 6
THEOREM 4 Let m be a positive integer. The integers a and b are congruent modulo m if and only if there is an integer k such that a = b + km. Example: 13 mod 5 43 mod 5 because 13 = 43 + (-6)(5)
THEOREM 4 EXAMPLES Is 13 and 67 congruent modulo 8? Is 37 and 64 congruent modulo 17? Is 123 and 51 congruent modulo 24?
THEOREM 5 Let m be a positive integer. If a b (mod m) and c d (mod m), then a + c b + d (mod m) and ac bd (mod m).
THEOREM 5 Example: 7 2 (mod 5) and 11 1(mod 5), it follows from Theorem 5 that 18 = 7 + 11 2 + 1 = 3 (mod 5)and that 77 = 7 11 2 1 = 2 (mod 5).
THEOREM 5 EXAMPLE 32 2 (mod 5) and 51 1(mod 5) 24 3 (mod 7) and 19 5(mod 7)
Arithmetic Modulo m We can define arithmetic operations on Zm, the set of nonnegative integers less than m, that is, the set {0, 1,..., m 1}. 1) Addition Modulo m 2) Multiplication Modulo m
Addition Modulo m In particular, we define addition of these integers, denoted by + m by a + m b = (a + b) mod m, where the addition on the right-hand side of this equation is the ordinary addition of integers,
Multiplication Modulo m and we define multiplication of these integers, denoted by m by a m b = (a b) mod m, where the multiplication on the righthand side of this equation is the ordinary multiplication of integers.
EXAMPLE Use the definition of addition and multiplication in Z m to find 7 + 11 9 and 7 11 9
HOMEWORK Give an example for each showing that arithmetic modulo m has the following properties: Closure Associativity Commutativity Identity elements Additive inverses Distributivity
SEATWORK 1. Does 17 divide each of these numbers? a) 68 b) 84 c) 357 d) 1001
SEATWORK 2. What are the quotient and remainder when a) 19 is divided by 7? b) 111 is divided by 11? c) 789 is divided by 23? e) 0 is divided by 19? f ) 3 is divided by 5? g) 1 is divided by 3?
SEATWORK 3. Suppose that a and b are integers, a 4 (mod 13), and b 9 (mod 13). Find the integer c with 0 c 12 such that a) c 9a (mod 13). b) c 11b (mod 13). c) c a + b (mod 13). d) c 2a + 3b (mod 13). e) c a 2 + b 2 (mod 13). f ) c a 3 b 3 (mod 13).
SEATWORK 4. Evaluate these quantities. a) 17 mod 2 b) 144 mod 7 c) 101 mod 13 d) 199 mod 19
SEATWORK 5. Find a div m and a mod m when a) a = 111, m = 99 b) a = 9999, m = 101 c) a = 10299, m = 999 d) a = 123456, m = 1001
SEATWORK 6. Find the integer a such that a) a 43 (mod 23) and 22 a 0. b) a 17 (mod 29) and 14 a 14. c) a 11 (mod 21) and 90 a 110.
SEATWORK 7. List five integers that are congruent to 4 modulo 12 8. Decide whether each of these integers is congruent to 3 modulo 7 a) 37 b) 66 c) 17 d) 67
SEATWORK 9. Find each of these values. a) (177 mod 31 + 270 mod 31) mod 31 b) (177 mod 31 270 mod 31) mod 31