# Euclids division algorithm

2
by vbhk

2014-05-14T11:42:26+05:30
Euclid's division algorithm is a technique to compute the hcf of two given positive integers.
( hcf of two positive integers a and b is the largest positive integer d that divides both a and b)

2014-05-14T11:56:33+05:30
The Euclidean algorithm proceeds in a series of steps such that the output of each step is used as an input for the next one. Let k be an integer that counts the steps of the algorithm, starting with zero. Thus, the initial step corresponds to k = 0, the next step corresponds to k = 1, and so on.Each step begins with two non negative remainders rk−1 and rk−2. Since the algorithm ensures that the remainders decrease steadily with every step, rk−1 is less than its predecessor rk−2. The goal of the kth step is to find a quotient qk and remainder rk such that the equation is satisfied
rk−2 = qk rk−1 + rk where rk < rk−1. In other words, multiples of the smaller number rk−1 are subtracted from the larger number rk−2 until the remainder is smaller than the rk−1.In the initial step (k = 0), the remainders r−2 and r−1 equal a and b, the numbers for which the GCD is sought. In the next step (k = 1), the remainders equal b and the remainder r0of the initial step, and so on. Thus, the algorithm can be written as a sequence of equations
a = q0 b + r0
b = q1 r0 + r1
r0 = q2 r1 + r2
r1 = q3 r2 + r3
If a is smaller than b, the first step of the algorithm swaps the numbers. For example, if a < b, the initial quotient q0 equals zero, and the remainder r0 is a. Thus, rk is smaller than its predecessor rk−1 for all k ≥ 0.Since the remainders decrease with every step but can never be negative, a remainder rN must eventually equal zero, at which point the algorithm stops. The final nonzero remainder rN−1 is the greatest common divisor of a and b. The number N cannot be infinite because there are only a finite number of nonnegative integers between the initial remainder r0 and zero.
Given positive integers a and b [a>b] there exist unique integers q and r satisfying a=bq+r, where 0 is less than or equal to r less than b.