I am having trouble understanding why the Euclidean algorithm for finding the GCD of two numbers always works?
I found some resources here (http://www.cut-the-knot.org/blue/Euclid.shtml), and here(http://sites.math.rutgers.edu/~greenfie/gs2004/euclid.html).
But I am a little confused about how they approach it here. I understand that if we have two numbers, a and b, then the greatest common divisor of a and b has to be less than a, and if a divides b, then a will have to be the GCD.
But I am confused about what happens when:
b=a*q+r
So now, we are saying that we take a/r, correct? Why should we do this at all?
Best Answer
I will try to explain
by using a standard argument in number theory: showing that a problem is equivalent to the same problem for smaller numbers.
Start with two numbers $a > b \ge 0$. You want to know two things:
their greatest common divisor $g$,
and how to represent $g$ as a combination of $a$ and $b$
It's clear that you know both of these things in the easy special case when $b = 0$.
Suppose $b > 0$. The divide $a$ by $b$ to get a quotient $q$ and a remainder $r$ strictly smaller than $b$: $$ a = bq + r. \quad \text{(*)} $$
Now any number that divides both $a$ and $b$ also divides $r$, so divides both $b$ and $r$. Also any number that divides both $b$ and $r$ also divides $a$, so divides both $a$ and $b$. That means that the greatest common divisor of $a$ and $b$ is the same as the greatest common divisor of $b$ and $r$, so (1) has the same answer $g$ for both those pairs.
Moreover, if you can write $g$ as a combination of $b$ and $r$ then you can write it as a combination of $a$ and $b$ (substitute in (*)). That means if you can solve (2) for the pair $(b,r)$ then you can solve it for the pair $(a,b)$.
Taken together, this argument shows that you can replace your problem for $(a,b)$ by the same problem for the smaller pair $(b,r)$. Since the problem can't keep getting smaller forever, eventually you will reach $(z, 0)$ and you're done.