数论-Euclidean algorithm

Description

Euclidean algorithm is used to find the gcd of two integers.
For two sorted integers a,b (a > b)

$$gcd(a,b) = gcd(a, a\ \mathbf{mod}\ b)$$

Proof

Step one: Left to Right

Let $a = k_1b + r$ and d is a common divisor of a and b
then

$$\frac{a}{d}=\frac{k_1b}{d} + \frac{r}{d}=m$$
Because of $$d|a \land d|b$$
$m$ is an integer, which implies $$d|r$$
i.e. d is a divisor of r
d is a common divisor of $a$ and $b\ \mathbf{mod}\ a$

Step two: Right to Left

Let $a = k_1b + r$ and $d$ is a common divisor of $a$ and $a\ \mathbf{mod}\ b$
From $a = k_1b + r$ we get

$$r = a - k_1b$$
then $$\frac{r}{d} = \frac{a}{d} - \frac{k_1b}{d}$$
Because of $$d|r \land d|a$$
then $$d|b$$

So the divisors of $a$ and $b$ are the divisors of $b$ and $a\ \mathbf{mod}\ b$.