Number theory is a branch of mathematics that deals with the properties and relationships of integers. It focuses on studying patterns and structures within numbers and their interactions.
A linear combination is a mathematical expression formed by multiplying each term with a constant and then adding them together. In the context of number theory, it refers to a combination of integers that can be expressed as a sum or difference of multiples of other integers.
The gcd of three numbers can be found by using the Euclidean Algorithm, which involves repeatedly dividing the larger number by the smaller number until the remainder is 0. The last non-zero remainder is the gcd.
This is because the gcd is the largest integer that can divide all three numbers a, b, and c without any remainder. Therefore, it can be expressed as a combination of a, b, and c multiplied by some constants.
By expressing the gcd as a linear combination of a, b, and c, we can prove that the gcd of any set of numbers can always be expressed as a combination of those numbers. This is a fundamental property of gcd and is useful in solving many mathematical problems.