About the Euclidean Algorithm
Purpose
The Euclidean algorithm finds the greatest common divisor of two integers. It works by repeated division. Each division replaces a pair with a smaller pair. The process ends when the remainder becomes zero. The last nonzero remainder is the gcd. This calculator also runs the extended method. It returns coefficients x and y. They satisfy ax + by = gcd(a,b).
Linear Combination Value
Linear combinations are useful in number theory. They show how a gcd is built from the original integers. This matters in modular arithmetic, Diophantine equations, and proof work. When the gcd is one, the numbers are relatively prime. Then one coefficient can become a modular inverse. That inverse helps solve congruences and cryptography exercises.
Advanced Input Handling
The tool accepts negative or positive integers. It computes the gcd from absolute values. Then it applies signs to the coefficients. The result still matches the original equation. You can also enter a target value. If the target is divisible by the gcd, the page scales the coefficients. This gives one solution for ax + by = target. A general solution is also shown with a free integer parameter.
Step Review
The step table shows every division line. It lists dividend, divisor, quotient, and remainder. This makes the work easy to check. The coefficient table gives the final Bezout identity. The lcm is included when possible. Download buttons save the same result for notes.
Study Use
Use the calculator when learning proofs. Use it when checking homework. Use it when a modular inverse is needed. Enter the two numbers first. Add a target only when needed. Press calculate. Read the result box before the form. Then study the steps below. The method is exact for integer inputs. Large numbers may create longer step tables, but the logic stays the same. Always verify the displayed equation. It confirms that the coefficients are correct.
Classroom Support
Because each quotient is recorded, you can compare manual work with the generated proof. The calculator does not hide the arithmetic. It exposes each remainder change and coefficient relation. This helps students see why the final identity is valid. Teachers can use the example table for class checks. Builders of math pages can reuse the layout and extend the inputs later. It keeps results organized without adding confusing screen clutter.