The principle of the Euclidean method

Simple proof of the principle of the Euclidean method

1.

Analytical:

8251=6105+2146, to be simple, I'm going to use A=b+c to show this.

So there are c=a-b then if there is d|a, and d|b, there must be d|a-b, that is, d|c, (D|a said: D is an approximate of a)

It can be seen that the number of conventions A and B must also be approximate to C.

Now suppose D is the greatest common divisor of a and B, then D must be an approximate number of C, so D is the B,c Convention,

Now it's time to prove that it's greatest common divisor:

2.

Prove:

Because of the a=b+c, so the b,c of the Convention is necessarily an approximate number of a, assuming (b,c) =e, ((b,c) =e said E is B and C greatest common divisor) then there is e|b+c, that is e|a according to "D is b,c convention number" know D|e, and because E

Also, the number of conventions of a, B, e|d, E=d visible (A, b) = (b,c) =d

(This idea is a generalization

,

It becomes the division of the law.)

15/6 The number must be less than 6, will become smaller, and eventually become greatest common divisor.

The principle of the Euclidean method