2.1 GCDs

GCDs I


Consider the two integers:
\(a = 21212121\)
\(b = 12121212\)

  1. What is the GCD of \(a\) and \(b\)?

    Hint: Looks scary, but it's not.

    We run the algorithm:

    \[GCD(21212121,12121212)\\ = GCD(12121212,9090909)\\ = GCD(9090909,3030303)\\ = GCD(3030303,0).\]

  2. How many steps of the Euclidean algorithm are needed to compute this GCD?

    A step of the Euclidean algorithm is defined as an application of the equation:

    \(GCD(x,y) = GCD(y, rem(x, y)).\)

    The algorithm begins with \((a,b)\) and ends with \((gcd(a,b),0)\).

    In the execution of the algorithm in Part 1 we needed 3 applications of the equation.