THEORY OF NUMBERS "Euclidean algorithm"

Euclidean algorithm is the algorithm to find the United Nations of two integers. Euclid, the inventor of the Euclidean algorithm is a Greek mathematician who wrote the algorithm in the famous book "Element".
Given two negative integers m and not n (m ≥ n). The following Euclidean algorithm to find the greatest common divisor of m and n.
Euclidean Algorithm 
1. If n = 0 then
m is the United Nations (m, n);
stop.
But if n ≠ 0
Proceed to step two:
2. Divide m by n and suppose that r is the remainder.
3. Change the value of m with the value and the value of n with r value, then reset back to step 1.
Example: m = 80, n = 12 and the filled condition m ≥ n

m = n. q + r
80 = 12.6 + 8
12 = 8. 1 + 4
8 = 4. 2 + 0
Time division last before 0 is 4, then the United Nations (80, 12) 4.
Theorem 1 (theorem Euclidean) such as m and n are two integers with the condition n> 0. If m divided by n then there are two unique integers q (quotient) and r (remainder), such that:

m = n. q + r
with 0 ≤ r ≤ n
example: (i) 1987 divided by 97 gives the results for the remaining 20 and 47.
1987 = 97. 20 + 47
(Ii) - 22 divided by 3 gives the results for - 8 and the remaining 2:
-22 = 3 (- 8) + 2
But - 22 = 3 (- 7) - 1 wrong, because r = - 1 are not eligible
0 ≤ r ≤ n.