Not necessarily true for an arbitrary linear combination, but there exists at least one linear combination equal to the gcd.MathSquareRoo said:Linear combination of u and v are equal to the gcd correct?
Certainly, gcd means greatest common divisor, so it's certainly a divisor.And the gcd divides u and v I believe.
MathSquareRoo said:So then au divides n and vb divides n?
The Euclidean algorithm is a method used to find the greatest common divisor (GCD) of two numbers. It is based on the principle that the GCD of two numbers is equal to the GCD of the smaller number and the remainder when the larger number is divided by the smaller number. It is named after the Greek mathematician Euclid.
The Euclidean algorithm works by repeatedly dividing the larger number by the smaller number and using the remainder as the new smaller number until the remainder is equal to 0. The last non-zero remainder is then the GCD of the two numbers.
The Euclidean algorithm is significant in number theory because it provides a quick and efficient method for finding the GCD of two numbers. It is also used in other areas of mathematics, such as cryptography and modular arithmetic.
Yes, the Euclidean algorithm can be used for any numbers that can be divided with remainders, such as rational numbers or polynomials. However, it is most commonly used for integers.
Yes, the Euclidean algorithm can be time-consuming for very large numbers. It also does not work for numbers that are not relatively prime (i.e. do not have a GCD of 1).