Greatest common divisor (gcd)

Definition

The greatest common divisor (gcd) of two or more integers is the largest positive integer that divides each of the numbers without leaving a remainder. This concept is crucial in simplifying fractions, factoring polynomials, and finding common factors, which all play an important role in algebraic operations on polynomials.

5 Must Know Facts For Your Next Test

1. The gcd can be calculated using the Euclidean algorithm, which involves a series of division steps.
2. For polynomials, the gcd can also refer to the highest degree polynomial that divides two or more polynomials without a remainder.
3. If one of the integers is zero, then the gcd is defined as the absolute value of the other integer.
4. The gcd is useful in simplifying rational expressions by canceling common factors in the numerator and denominator.
5. For any integers a and b, it holds that gcd(a, b) × lcm(a, b) = |a × b|, where lcm stands for least common multiple.

Review Questions

• How can you apply the Euclidean algorithm to find the gcd of two given integers?
  • To find the gcd of two integers using the Euclidean algorithm, start with your two numbers, let's say a and b. Divide a by b and take note of the remainder r. Then replace a with b and b with r. Repeat this process until b becomes zero. The non-zero value of a at this point is the gcd. This method works because it systematically reduces the problem while maintaining equivalence to the original gcd.
• In what way does finding the gcd of polynomials differ from finding the gcd of integers?
  • Finding the gcd of polynomials involves identifying the highest degree polynomial that divides both polynomials without leaving a remainder. This requires methods such as polynomial long division or synthetic division. While integer gcd relies on numerical division and properties like prime factorization, polynomial gcd incorporates concepts from algebra such as degree comparison and variable coefficients, making it a bit more complex.
• Evaluate how understanding gcd can impact your ability to work with algebraic expressions and equations.
  • Understanding the concept of gcd significantly enhances your ability to simplify algebraic expressions and solve equations. When you can identify and factor out common divisors, you streamline calculations and reduce complexity in your work. This ability not only aids in simplifying fractions but also helps in solving polynomial equations efficiently by identifying roots or factoring expressions into solvable parts. Mastery of gcd thus lays foundational skills for advanced algebraic techniques.

Related terms

Euclidean Algorithm: A method for finding the gcd of two integers by repeatedly applying the principle that gcd(a, b) = gcd(b, a mod b) until one of the numbers becomes zero.

Polynomial Factorization: The process of breaking down a polynomial into simpler polynomials that, when multiplied together, give back the original polynomial.

Common Factor: A number that divides two or more numbers without leaving a remainder; the gcd is the largest among them.