7.3 Euclid’s Algorithm

Euclid’s algorithm for greatest common divisor

${ gcd(b,amod b) if b > 0 gcd(a,b) = a if b = 0$
gcd(a,b) = gcd(b,a)
gcd(a,b) = gcd(-a,b)
At most k recursive calls if a > b > 0 and b < F_k+1
The limit is reached for Fibonacci numbers
gcd(F_k+1,F_k) = gcd(F_k, (F_k+1 mod F_k)) = gcd(F_k,F_k-1)
The limit on the number of calls is of order O(log F_k)

$( V~ - V~ ) F = V~ 1 (1+--5)k- (1---5)k k 5 V ~ 2 2 ~~ V~ 1-(1+-5)k 5 2$
Least common multiple
lcm(x,y) = xy/gcd(x,y)
There exists a related algorithm to solve modular linear equations $ax = b(mod n)$ for x