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Questions and Answers
What is the definition of a prime number?
Which statement correctly defines the Greatest Common Divisor (GCD)?
Which of these is true according to Euclid's Lemma?
What characterizes a quadratic equation?
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Which method is employed to solve quadratic equations?
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What does the Fundamental Theorem of Arithmetic state?
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Which of the following represents a linear equation?
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In the context of number theory, what is a composite number?
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What is the purpose of the Euclidean algorithm?
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Which of the following statements about algebraic equations is NOT true?
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Study Notes
Number Theory
 Definition: Number theory is the branch of mathematics that deals with the properties and behavior of integers and other whole numbers.

Key concepts:
 Divisibility: a  b if a divides b exactly without leaving a remainder
 Prime numbers: positive integers greater than 1 that are divisible only by 1 and themselves
 Composite numbers: positive integers greater than 1 that are not prime
 Greatest Common Divisor (GCD): the largest number that divides both a and b exactly
 Euclidean algorithm: a method for finding the GCD of two numbers

Theorems:
 Fundamental Theorem of Arithmetic: every positive integer can be expressed as a product of prime numbers in a unique way
 Euclid's Lemma: if p is prime and p  ab, then p  a or p  b
Algebraic Equations
 Definition: An algebraic equation is an equation that involves variables and constants, and can be expressed using only addition, subtraction, multiplication, and division, and roots (such as square roots or cube roots).

Types of equations:
 Linear equations: equations of the form ax + by = c, where a, b, and c are constants
 Quadratic equations: equations of the form ax^2 + bx + c = 0, where a, b, and c are constants
 Polynomial equations: equations of the form a_n x^n + a_(n1) x^(n1) + ... + a_1 x + a_0 = 0, where a_n, ..., a_1, a_0 are constants

Methods for solving equations:
 Factoring: expressing an equation as a product of simpler equations
 Quadratic formula: x = (b ± √(b^2  4ac)) / 2a, for solving quadratic equations
 Synthetic division: a method for dividing a polynomial by another polynomial

Applications:
 Solving systems of equations
 Finding roots of polynomials
 Modeling realworld problems
Number Theory
 Branch of mathematics focused on integers and their properties
 Divisibility: Indicated as a  b; means a divides b without remainder
 Prime Numbers: Greater than 1, only divisible by 1 and itself; examples include 2, 3, 5, 7
 Composite Numbers: Greater than 1 and not prime; includes numbers like 4, 6, 8
 Greatest Common Divisor (GCD): Largest integer that divides two numbers without leaving a remainder
 Euclidean Algorithm: A systematic method for calculating the GCD of two integers
 Fundamental Theorem of Arithmetic: States every positive integer can be uniquely expressed as a product of prime numbers
 Euclid's Lemma: If a prime number p divides the product of two integers ab, then p must divide at least one of a or b
Algebraic Equations
 Definition: Involves variables and constants, expressible through basic arithmetic operations and roots

Types:
 Linear Equations: Form ax + by = c; a, b, and c are constants, represents a straight line on a graph
 Quadratic Equations: Form ax² + bx + c = 0; represents a parabolic curve, solutions can be found using the quadratic formula
 Polynomial Equations: General form a_n x^n + a_(n1) x^(n1) +...+ a_1 x + a_0 = 0; includes multiple degrees of x

Methods for Solving Equations:
 Factoring: Process of breaking down an expression into simpler components that multiply to the original equation
 Quadratic Formula: x = (b ± √(b²  4ac)) / 2a; used for finding roots of quadratic equations
 Synthetic Division: Simplifies the process of dividing a polynomial by a linear factor

Applications:
 Solving systems of equations to find variable values that satisfy multiple conditions
 Finding roots of polynomials, relevant in calculus and function analysis
 Modeling realworld problems in fields like physics, finance, and engineering through algebraic relationships
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Description
Test your understanding of key concepts in number theory, including divisibility, prime numbers, composite numbers, and greatest common divisors.