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mod arithmetic a mod m is the remainder of a divided by m a mod m is the integer r such that a = qm + r and 0 <= r < m again, r is positive Examples 17 mod 3 = 2 17 mod 12 = 5 (5 o’clock) -17 mod 3 = 1
congruences a is congruent to b modulo m if m divides a - b
a is congruent to b mod m if and only if the remainder of adivided by m is equal to the remainder of b divided by m. proof
If a is congruent to b mod m and c is congruent to d mod mthen a+c is congruent to b+d mod m proof
If a is congruent to b mod m and c is congruent to d mod mthen ac is congruent to bd mod m proof
Mod arithmetic examples -133 mod 9 = 2 (but in Claire?) list 5 numbers that are congruent to 4 modulo 12 hash function h(k) = k mod 101 h( ) h( ) h( ) h( )
Chinese Remainder Theorem. How many people What is x? Divided into 4s: remainder 3 x ≡ 3 (mod 4) Divided into 5s: remainder 4 x ≡ 4 (mod 5) Chinese Remainder.
Chapter 13 Mathematic Structures 13.1 Modular Arithmetic Definition 1 (modulo). Let a be an integer and m be a positive integer. We denoted by a mod m.
Arithmetic Expressions Addition (+) Subtraction (-) Multiplication (*) Division (/) –Integer –Real Number Mod Operator (%) Same as regular Depends on the.
Factor A factor of an integer is any integer that divides the given integer with no remainder.
Divisibility Rules and Finding Factors
Greatest Common Factor and Least Common Multiples GCF and LCM.
Prime and Composite Numbers
Doing math In java.
A number that divides evenly into a larger number without a remainder Factor- Lesson 7 - Attachment A.
This will all add up in the end. Assignment operator =Simple Assignment operator Arithmetic Operators +Additive operator – Subtraction operator * Multiplication.
Multiplying and Dividing Integers When you MULTIPLY: Two positives equal a positive Two negatives equal a positive One positive & one negative equal.
Lesson 11-2 Remainder & Factor Theorems Objectives Students will: Use synthetic division and the remainder theorem to find P(r) Determine whether a given.
Congruence class arithmetic. Definitions: a ≡ b mod m iff a mod m = b mod m. a [b] iff a ≡ b mod m.
Chapter 4 Finite Fields. Introduction of increasing importance in cryptography –AES, Elliptic Curve, IDEA, Public Key concern operations on “numbers”
Division & Divisibility. a divides b if a is not zero there is a m such that a.m = b a is a factor of b b is a multiple of a a|b Division.
Prep Math Competition, Lec. 1Peter Burkhardt1 Number Theory Lecture 1 Divisibility and Modular Arithmetic (Congruences)
Arithmetic OperatorOperationExample +additionx + y -subtractionx - y *multiplicationx * y /divisionx / y Mathematical FormulaC Expressions b 2 – 4acb *
Factors are numbers you can multiply together to get another number Example: 2 and 3 are factors of 6, because 2 × 3 = 6 Objectives: SWBAT 1) find the.
Cyclic Groups. Definition G is a cyclic group if G = for some a in G.
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