Euclidean algorithm: Difference between revisions

The Euclidean algorithm (also called Euclid's algorithm) is an algorithm to determine the greatest common divisor (gcd) of two integers. It is one of the oldest algorithms known, since it appeared in Euclid's Elements around 300 BC. The algorithm does not require factoring.

Algorithm and implementation

Given two natural numbers a and b, first check if b is zero. If yes, then the gcd is a. If no, calculate c, the remainder after the division of a by b. Replace a with b, b with c, and start the process again.

The algorithm can be formulated in the Python programming language as follows:

   def gcd(a, b):
   while b != 0:
      a, b = b, a % b
   return abs(a)

For example, the gcd of 1071 and 1029 is computed by this algorithm to be 21 with the following steps:

a	b	c
1071	1029	42
1029	42	21
42	21	0
21	0

(The absolute value is used in the last line to ensure that the algorithm correctly deals with negative inputs; e.g. gcd(−7, 0) = 7.)

By keeping track of the quotients occurring during the algorithm, one can also determine integers p and q with ap + bq = gcd(a, b). This is known as the extended Euclidean algorithm.

These algorithms can be used in any context where division with remainder is possible. This includes rings of polynomials over a field as well as the ring of Gaussian integers, and in general all Euclidean domains.

Euclid originally formulated the problem geometrically, as the problem of finding a common "measure" for two line lengths, and his algorithm proceeded by repeated subtraction of the shorter from the longer segment. This is illustrated with the following implementation in Python, which works only for positive inputs and is considerably less efficient than the method explained above:

   def gcd(a, b):
   while a != b:
       if a > b:
           a = a - b
       else:
           b = b - a
   return a

Proof of correctness

The proof of this algorithm is not difficult. Suppose a and b are the numbers whose gcd has to be determined. And suppose the remainder of the division of a by b is c. Therefore a = qb + c where q is the quotient of the division. Now any common divisor of a and b also divides c (since c can be written as c = a − qb); similarly, any common divisor of b and c will also divide a. Thus the greatest common divisor of a and b is the same as the greatest comon divisor of b and c. Therefore it is enough if we continue the process with the numbers b and c. Since c is smaller in absolute value than b, we will reach c = 0 after finitely many steps.

Running time

When analyzing the runtime of Euclid's algorithm, it turns out that the inputs requiring the most divisions are two successive Fibonacci numbers, and the worst case requires Θ(n) divisions, where n is the number of digits in the input (see Big O notation). However, it must be noted that the divisions themselves are not atomic operations, since the size of the operands could be as large as n digits. The actual running time is therefore O(n²).

Euclid's algorithm is widely used in practice, especially for small numbers, due to its simplicity. An alternative algorithm exploiting the binary nature of modern digital computers might be slightly faster, although it too is O(n²).

Continued fractions

The quotients that appear when the Euclidean algorithm is applied to the inputs a and b are precisely the numbers occurring in the continued fraction representation of a/b. Take for instance the example of a = 1071 and b = 1029 used above. Here is the calculation with highlighted quotients:

1071 = 1029 × 1 + 42

1029 = 42 × 24 + 21

42 = 21 × 2 + 0

From this, one can read off that

${\frac {1071}{1029}}=\mathbf {1} +{\frac {1}{\mathbf {24} +{\frac {1}{\mathbf {2} }}}}$.

This method can even be used for real inputs a and b; if a/b is irrational, then the Euclidean algorithm won't terminate, but the computed sequence of quotients still represents the (now infinite) continued fraction representation of a/b.

See also: Least common multiple, Extended Euclidean algorithm

External links

• Euclid's Algorithm: Algorithm, generalization, game and related topics