Greatest Common Divisors

Let $a$ and $b$ be two integers. The greatest common divisor of $a$ and $b$ is the biggest number that divides both of them. We denote it by ``$\gcd(a,b)$''. Thus,

Definition 2.1  

$$\gcd(a,b)=\max\{d : d \mid a$$ and $$d\mid b\}.$$

Warning: We define $\gcd(0,0)=0$, instead of ``infinity''.

Here are a few gcd's:

$$\gcd(1,2)=1,\quad \gcd(0,a)=\gcd(a,0)=a, \quad\gcd(3,27)=3, \quad\gcd(2261,1275)=?$$

Warning: In Davenport's book, he denotes our $\gcd$ by HCF and calls it the ``highest common factor''. I will use the notation $\gcd$ because it is much more common.