# gcd, linear combination problem

• Sep 14th 2010, 04:34 PM
Janu42
gcd, linear combination problem
For integers a and b, what is the relationship between gcd(a,b) and the set of integer linear combinations of a and b?
• Sep 14th 2010, 05:23 PM
Chris11
gcd(a,b) divides all linear combos of a and b.
• Sep 15th 2010, 07:44 AM
melese
Quote:

Originally Posted by Janu42
For integers a and b, what is the relationship between gcd(a,b) and the set of integer linear combinations of a and b?

Suppose $\displaystyle a$ and $\displaystyle b$ are integers - not both zero, and let $\displaystyle L=\left\{ax+by|x,y\in\mathbb{Z}\right\}$ be the set of all linear combinations of $\displaystyle a$ and $\displaystyle b$.

The following relationship holds: Every linear combination of $\displaystyle a$ and $\displaystyle b$ (a member in $\displaystyle L$) is a multiple of $\displaystyle gcd(a,b)$; conversely, any multiple of $\displaystyle gcd(a,b)$ is a linear combination of $\displaystyle a$ and $\displaystyle b$.
In short, $\displaystyle L$ is precisely the set of all multiples of $\displaystyle gcd(a,b)$.

BTW, the greatest common divisor of $\displaystyle a$ and $\displaystyle b$ is the smallest postive linear combination of $\displaystyle a$ and $\displaystyle b$.