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HomeHomework HelpmathematicsGreatest Common Divisor

Greatest Common Divisor

The Greatest Common Divisor (GCD) of two integers is the largest positive integer that divides both numbers without leaving a remainder. In the context of polynomials, division is characterized by the existence of unique polynomials Q and R such that A = BQ + R, where R is either the zero polynomial or has a degree less than that of B.

intermediate
2 hours
Mathematics
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Overview

The Greatest Common Divisor (GCD) is a fundamental concept in mathematics that helps in simplifying fractions and solving problems involving divisibility. It can be calculated using the Euclidean algorithm, which is efficient and straightforward. Understanding GCD is essential for various applicatio...

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Key Terms

Divisibility
A number a is divisible by b if there is an integer c such that a = b * c.

Example: 10 is divisible by 2 because 10 = 2 * 5.

GCD
The greatest common divisor of two or more integers is the largest integer that divides them without a remainder.

Example: GCD of 8 and 12 is 4.

Euclidean Algorithm
A method for finding the GCD of two numbers by repeated division.

Example: To find GCD(48, 18), divide 48 by 18, then 18 by the remainder.

Polynomial
An expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication.

Example: 3x² + 2x + 1 is a polynomial.

Degree of a Polynomial
The highest power of the variable in a polynomial.

Example: In 4x³ + 2x², the degree is 3.

Long Division
A method for dividing polynomials similar to long division with numbers.

Example: Dividing x² + 5x + 6 by x + 2.

Related Topics

Prime Factorization
Breaking down numbers into their prime factors, which helps in finding GCD.
intermediate
Linear Algebra
Study of vectors and matrices, which often involves polynomial equations.
advanced
Number Theory
Branch of mathematics dealing with integers and their properties, including GCD.
advanced

Key Concepts

DivisibilityEuclidean AlgorithmPolynomial DivisionFactoring