Difference between revisions of "Greatest common divisor"

Revision as of 10:23, 19 April 2008

The greatest common divisor (GCD, or GCF (greatest common factor)) of two or more integers is the largest integer that is a divisor of all the given numbers.

The GCD is sometimes called the greatest common factor (GCF).

A very useful property of the GCD is that it can be represented as a sum of the given numbers with integer coefficients. From here it immediately follows that the greatest common divisor of several numbers is divisible by any other common divisor of these numbers.

Finding the GCD

Using prime factorization

Once the prime factorizations of the given numbers have been found, the greatest common divisor is the product of all common factors of the numbers.

Example:

$270=2\cdot3^3\cdot5$ and $144=2^4\cdot3^2$ . The common factors are 2 and $3^2$ , so $GCD(720,144)=2\cdot3^2=18$ .

Euclidean algorithm

The Euclidean algorithm is much faster and can be used to give the GCD of any two numbers without knowing their prime factorizations. To find the greatest common divisor of more than two numbers, one can use the recursive formula $GCD(a_1,\dots,a_n)=GCD(GCD(a_1,\dots,a_{n-1}),a_n)$ .

Using least common multiple

The GCD of two numbers can also be found using the equation $GCD(x, y) \cdot LCM(x, y) = x \cdot y$ , where $LCM(x,y)$ is the least common multiple of $x$ and $y$ .

Template:Wikify

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-The '''greatest common divisor''' ('''GCD''') of two or more [[integers]] is the largest integer that is a [[divisor]] of all the given numbers.
+The '''greatest common divisor''' ('''GCD''', or '''GCF''' ('''greatest common factor''')) of two or more [[integer]]s is the largest integer that is a [[divisor]] of all the given numbers.
 The GCD is sometimes called the '''greatest common factor''' ('''GCF''').
@@ Line 5: / Line 5: @@
 A very useful property of the GCD is that it can be represented as a sum of the given numbers with integer coefficients. From here it immediately follows that the greatest common divisor of several numbers is divisible by any other common divisor of these numbers.
+== Finding the GCD ==
+=== Using prime factorization ===
+Once the [[prime factorization]]s of the given numbers have been found, the greatest common divisor is the product of all common factors of the numbers.
+Example:
-The GCD can be found in two ways.  The first way involves factoring the numbers, and the second way uses [[Euclidean algorithm]].
+<math>270=2\cdot3^3\cdot5</math> and <math>144=2^4\cdot3^2</math>. The common factors are 2 and <math>3^2</math>, so <math>GCD(720,144)=2\cdot3^2=18</math>.
-----
+=== Euclidean algorithm ===
+The [[Euclidean algorithm]] is much faster and can be used to give the GCD of any two numbers without knowing their prime factorizations. To find the greatest common divisor of more than two numbers, one can use the recursive formula <math>GCD(a_1,\dots,a_n)=GCD(GCD(a_1,\dots,a_{n-1}),a_n)</math>.
-Once the [[prime factorization]]s of the given numbers has been found, the greatest common divisor is the product of all common factors of the numbers.
+=== Using least common multiple ===
+The GCD of two numbers can also be found using the equation <math>GCD(x, y) \cdot LCM(x, y) = x \cdot y</math>, where <math>LCM(x,y)</math> is the [[least common multiple]] of <math>x</math> and <math>y</math>.
-Example:<br>
-<math>270=2\times3^3\times5</math><br>
-<math>144=2^4\times3^2</math>
-The common factors are 2 and <math>3^2</math>, so the GCD is <math>2\times3^2=18</math>
-Another Example:<br>
-<math>1200=2^4\times3\times5</math><br>
-<math>720=2^4\times3^2\times5</math><br>
-<math>288=2^5\times3^2</math>
-The common factors are <math>2^4</math> and 3, making the GCD <math>{2^4\times3=48}</math>.
-----
-The Euclidean algorithm is much faster and can be used to give the GCD of any two numbers without knowing their prime factorizations. To find the greatest common divisor of more than two numbers, one can use the recursive formula <math>GCD(a_1,\dots,a_n)=GCD(GCD(a_1,\dots,a_{n-1}),a_n)</math>.
-----
-Another useful trick is the equation
-<math>GCD(x, y) \cdot LCM(x, y) = x \cdot y</math>
-Using this, you can find the GCD rather quickly if you know the LCM and the two numbers.  This equation is also useful for finding one of the numbers if you know the other number, the GCD and the LCM.
 {{wikify}}
 [[Category:Number theory]]

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